qid stringlengths 1 7 | Q stringlengths 87 7.22k | dup_qid stringlengths 1 7 | Q_dup stringlengths 97 10.5k |
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77550 | Prove that [imath]\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0[/imath], [imath]x \in \Bbb R[/imath].
Why is [imath]\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}[/imath] Can we generalize it to any exponent [imath]x \in \Bbb R[/imath]? This is to say, is [imath]\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}[/imath]... | 352360 | How to prove that [imath]\lim_{n\to\infty}\frac{a^n}{n!}=0[/imath]?
What I have got: For any [imath]a\in\mathbb{R}[/imath], we can find an [imath]N[/imath], [imath]N\gt a[/imath], such that, [imath]\lim_{n\to\infty}\frac{a^n}{n!}=\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdot\ldots\cdot\frac{a}{N-1}\cdot\frac{a}{N}\... |
37647 | If [imath]\sum a_n b_n <\infty[/imath] for all [imath](b_n)\in \ell^2[/imath] then [imath](a_n) \in \ell^2[/imath]
I'm trying to prove the following: If [imath](a_n)[/imath] is a sequence of positive numbers such that [imath]\sum_{n=1}^\infty a_n b_n<\infty[/imath] for all sequences of positive numbers [imath](b_n)[/... | 359833 | Prove that the sequence is in [imath]\ell^{2}[/imath].
Let [imath](a_{n})[/imath] be a sequence of complex numbers such that for every [imath](b_{n})\in \ell^{2}[/imath]the series [imath]\sum_{1}^{\infty}a_{n}b_{n}[/imath] converges. Prove that [imath](a_{n})\in \ell^{2}.[/imath] What I've tried so far is Let [imath]T... |
191165 | Proving an equality involving compositions of an integer
Let's consider various representations of a natural number [imath]n \geq 4[/imath] as a sum of positive integers, in which the order of summands is important (i.e. compositions). The task is to prove the number [imath]3[/imath] appears altogether [imath]n2^{n-5... | 306541 | Number of times the integer [imath]3[/imath] occurs in all [imath]2^{n-1}[/imath] compositions of [imath]n[/imath]
Suppose [imath]n\ge4[/imath]. Show that in a list of all [imath]2^{n-1}[/imath] compositions of [imath]n[/imath], the integer [imath]3[/imath] occurs exactly [imath]n2^{n-5}[/imath] times. [Hint: Look at ... |
60590 | Category-theoretic limit related to topological limit?
Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a category-theoretic setting of some non-trivial topological space such... | 420753 | Relation between inverse limits (and direct limits) with limits in calculus.
What is the relation between inverse limit (and direct limit) with limits in calculus? Are there some special cases that an inverse limit (or direct limit) is a limit in calculus (for example, the limit of a sequence [imath]a_0, a_1, \ldots[/... |
47009 | Proof of [imath]\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}[/imath]
It's a standard result that given [imath]X_1,\cdots ,X_n [/imath] random sample from [imath]N(\mu,\sigma^2)[/imath], the random variable [imath]\frac{(n-1)S^2}{\sigma^2}[/imath] has a chi-square distribution with [imath](n-1)[/imath] degrees of f... | 1773969 | Deriving [imath](n-1)\frac{S^2}{\sigma^2} \sim \chi^2(n-1)[/imath]
I can accept the fact that [imath]Z^2 = \dfrac{\left(X-\mu\right)^2}{\sigma^2} \sim \chi^2(1)[/imath] without knowing too much about this mysterious [imath]\chi[/imath]-function, but I'm wondering how I can show that [imath](n-1)\dfrac{S^2}{\sigma^2} \... |
41303 | Examples and further results about the order of the product of two elements in a group
Let [imath]G[/imath] be a group and let [imath]a,b[/imath] be two elements of [imath]G[/imath]. What can we say about the order of their product [imath]ab[/imath]? Wikipedia says "not much": There is no general formula relating the... | 314850 | Example of a group where [imath]o(a)[/imath] and [imath]o(b)[/imath] are finite but [imath]o(ab)[/imath] is infinite
Let G be a group and [imath]a,b \in G[/imath]. It is given that [imath]o(a)[/imath] and [imath]o(b)[/imath] are finite. Can you give an example of a group where [imath]o(ab)[/imath] is infinite? |
117998 | Finding the error in this proof that 1=2
I have a "proof" that has an error in it and my goal is to figure out what this error is. The proof: If [imath]x = y[/imath], then [imath] \begin{eqnarray} x^2 &=& xy \nonumber \\ x^2 - y^2 &=& xy - y^2 \nonumber \\ (x + y)(x - y) &=& y(x-y) \nonumber \\ x + y &=& y \nonumber \... | 389180 | Where is wrong in this proof
Suppose [imath]a=b[/imath]. Multiplying by [imath]a[/imath] on both sides gives [imath]a^2 = ab[/imath]. Then we subtract [imath]b^2[/imath] on both sides, and get [imath]a^2-b^2 = ab-b^2[/imath]. Obviously, [imath](a-b)(a+b) = b(a-b)[/imath], so dividing by [imath]a - b[/imath], we find [... |
93453 | How to prove that [imath]\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}][/imath], for [imath]p_i[/imath] prime?
This is Exercise 18.14 from Algebra, Isaacs. [imath]p_{1}\ ,\ p_{2}\ ,\ ... p_{n}[/imath] are different prime numbers. How to show that [imath]... | 1096809 | Is [imath]\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})[/imath].
Is [imath]\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)[/imath]? Say [imath]L=\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)[/imath] and [imath]K=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)[/imath]. It is easy to show t... |
27539 | Proving a special case of the binomial theorem: [imath]\sum^{k}_{m=0}\binom{k}{m} = 2^k[/imath]
I want to know if I can get some help with this proof. I tried, but I just cannot seem to get [imath]2^{k}[/imath]. It states that, For [imath]k \in \mathbb{Z}_{\ge 0}[/imath], [imath]\sum^{k}_{m=0}\binom{k}{m} = 2^k[/imat... | 177405 | Prove by induction: [imath]2^n = C(n,0) + C(n,1) + \cdots + C(n,n)[/imath]
This is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. [imath]2^n = C(n,0) + C(n,1) + \cdots + C(n,n).[/imath] Prove this statement is true for all [imath]n \ge 0[/imath] by induction. |
30111 | Unique quadratic subfield of [imath]\mathbb{Q}(\zeta_p)[/imath] is [imath]\mathbb{Q}(\sqrt{p})[/imath] if [imath]p \equiv 1[/imath] [imath](4)[/imath], and [imath]\mathbb{Q}(\sqrt{-p})[/imath] if [imath]p \equiv 3[/imath] [imath](4)[/imath]
I want to prove the assertion: The unique quadratic subfield of [imath]\math... | 2812412 | [imath]\xi =e^{2*\pi/pi }[/imath] , proof that is only two quadratic extention to [imath]\Bbb Q [/imath] that contained in [imath]Q(\xi_p)[/imath]
[imath]\xi_p =e^{2\pi/p*i }[/imath] , proof that is only two quadratic extention to [imath]\Bbb Q [/imath] that contained in [imath]Q(\xi_p)[/imath] such that [imath]\Bbb Q... |
134714 | Does [imath]|x|^p[/imath] with [imath]0 satisfy the triangle inequality on \mathbb{R}?[/imath]
I am curious about whether [imath]|x|^p[/imath] with [imath]0<p<1[/imath] satisfy [imath]|x+y|^p\leq|x|^p+|y|^p[/imath] for [imath]x,y\in\mathbb{R}[/imath]. So far my trials show that this seems to be right... So can anybod... | 1763343 | Root distance function in Metric space
Let [imath]\mathbf X = \Bbb R[/imath] with distance function defined by [imath]d(x,y) = {|x-y|}^\alpha[/imath] , where [imath]\alpha \in \Bbb R[/imath] [imath](0<\alpha\le1)[/imath]. Prove that [imath](\Bbb R , d)[/imath] is a metric space. The first three properties are easy, so... |
182888 | Limit of [imath]\frac{1}{x} - \frac{1}{\sin{(x)}}[/imath]
Prove, without using l'Hôpital's Rule, that [imath]\lim\limits_{x \to 0}{\dfrac{1}{x} - \dfrac{1}{\sin{(x)}}} = 0[/imath]. I proved that there exists a [imath]s >0[/imath] such that [imath]\forall x \in (-s,s)[/imath] [imath]\Rightarrow[/imath] [imath]\dfrac{1... | 94864 | What is the result of [imath]\lim\limits_{x \to 0}(1/x - 1/\sin x)[/imath]?
Find the limit: [imath]\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)[/imath] I am not able to find it because I don't know how to prove or disprove [imath]0[/imath] is the answer. |
282045 | Three linked question on non-negative definite matrices.
1.a symmetric matrix in [imath]\mathbb{M}_n(\mathbb{R})[/imath] is said to be non-negative definite if [imath]x^Tax≥0[/imath] for all (column) vectors [imath]x\in \mathbb{R}^n[/imath]. Which of the following statements are true? (a) If a real symmetric [imath]n\... | 311686 | a multiple choice question on non-negative definite matrices
A symmetric matrix in [imath]\mathbb{M}_n(\mathbb{R})[/imath] is said to be non-negative definite if [imath]x^TAx≥0[/imath] for all (column) vectors [imath]x \in \mathbb{R}^n[/imath]. Which of the following statements are true? a. If a real symmetric [imath]... |
9776 | How to raise a complex number to the power of another complex number?
How do I calculate the outcome of taking one complex number to the power of another, ie [imath]\displaystyle {(a + bi)}^{(c + di)}[/imath]? | 1027646 | How to calculate [imath]i^i[/imath]
I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python a**b = [imath]a^b[/imath] ) and found out the answer to be a real number with value [imath]0.2079[/imath], How to calculate this value of [imath]i^i[/imath... |
101157 | A commutative ring is a field iff the only ideals are [imath](0)[/imath] and [imath](1)[/imath]
Let [imath]R[/imath] be a commutative ring with identity. Show that [imath]R[/imath] is a field if and only if the only ideals of [imath]R[/imath] are [imath]R[/imath] itself and the zero ideal [imath](0)[/imath]. I can't f... | 504192 | Every field has exactly 2 ideals
I want to prove that every field has exactly 2 ideals. I know that the two ideals are [imath]0[/imath] and the whole set itself. That is clear to me. I am a little unsure how to prove it though. |
35598 | Why are addition and multiplication commutative, but not exponentiation?
We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that can be used to define these operators. One of th... | 2077612 | Why do we lose the abelian property as soon as we reach exponentiation?
The first operation, addition, is abelian and so is multiplication. However, the next operation, exponentiation is not! Why is this? I understand that [imath]2^3[/imath] and [imath]3^2[/imath] are not equal but why do we suddenly lose this propert... |
214871 | Any subgroup of index [imath]p[/imath] in a [imath]p[/imath]-group is normal.
Let [imath]p[/imath] be a prime number and [imath]G[/imath] a finite group where [imath]|G|=p^n[/imath], [imath]n \in \mathbb{Z_+}[/imath]. Show that any subgroup of index [imath]p[/imath] in it is normal in [imath]G[/imath]. Conclude that a... | 1960875 | Every subgroup of index [imath]p[/imath] is normal in [imath]G[/imath].
Could anyone give me a hint on how to solve the following problem? Prove that if [imath]p[/imath] is a prime and [imath]G[/imath] is a group of order [imath]p^{\alpha}[/imath] for some [imath]\alpha \in \mathbb{Z}^{+}[/imath], then every subroup ... |
5248 | Evaluating the integral [imath]\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2[/imath]?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: [imath]$$\displaystyle\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$[/imath] Well, can ... | 412847 | How can I evaluate [imath]\int_0^\infty \frac{\sin x}{x} \,dx[/imath]? [may be duplicated]
How can I evaluate [imath]\displaystyle\int_0^\infty \frac{\sin x}{x} \, dx[/imath]? (Let [imath]\displaystyle \frac{\sin0}{0}=1[/imath].) I proved that this integral exists by Cauchy's sequence. However I can't evaluate what is... |
9188 | Is [imath]\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})[/imath]?
In this post we saw isomorphism of vector spaces over [imath]\mathbb{Q}[/imath]. Just came across this question: Is [imath]\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})[/imath]? I know these as [imath]\mathbb{Q}[/imath]-Vector spaces, are isomorphi... | 759113 | How to show that [imath]\mathbb Q(\sqrt 2)[/imath] is not field isomorphic to [imath]\mathbb Q(\sqrt 3).[/imath]
How to show that [imath]\mathbb Q(\sqrt 2)[/imath] is not field isomorphic to [imath]\mathbb Q(\sqrt 3)?[/imath] My text provides the hint as: Any isomorphism from [imath]\mathbb Q(\sqrt 2)\to\mathbb Q(\sq... |
438 | Why [imath]\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2[/imath]?
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\ \\ \sqrt{\frac1{-1}} &= \frac1i \\ \\ \sqrt{\frac{-1}1} &=... | 351929 | Square and square root and negative numbers
Are they equal? -5 = [imath]\sqrt{(-5)^2}[/imath] |
49169 | Why [imath]\sqrt{-1 \times -1} \neq \sqrt{-1}^2[/imath]?
We know [imath]i^2=-1 [/imath]then why does this happen? [imath] i^2 = \sqrt{-1}\times\sqrt{-1} [/imath] [imath] =\sqrt{-1\times-1} [/imath] [imath] =\sqrt{1} [/imath] [imath] = 1 [/imath] EDIT: I see this has been dealt with before but at least with this answer... | 755970 | Value of [imath]i^2[/imath] in complex numbers
Please solve this doubt : we know that [imath]\sqrt{a}\sqrt{b}=\sqrt{ab}[/imath] and [imath]i^2 = -1[/imath]. But [imath]i= \sqrt{-1}[/imath] which implies that [imath]i^2 = i \cdot i = \sqrt{-1}\sqrt{-1} = \sqrt{1} = 1[/imath] that is [imath]i^2 = 1[/imath]. So what is ... |
33215 | What is 48÷2(9+3)?
There is a huge debate on the internet on [imath]48÷2(9+3)[/imath]. I figured if i wanted to know the answer this is the best place to ask. I believe it is 2 as i believe it is part of the bracket operation in BEDMAS. http://www.mathway.com/ agrees with me. I also said if [imath]48÷2*(9+3)[/imath] w... | 16502 | Do values attached to integers have implicit parentheses?
Given [imath]5x/30x^2[/imath] I was wondering which is the correct equivalent form. According to BEDMAS this expression is equivalent to [imath]5*\cfrac{x}{30}*x^2[/imath] but, intuitively, I believe that it could also look like: [imath]\cfrac{5x}{30x^2}[/imath... |
21330 | "Closed" form for [imath]\sum \frac{1}{n^n}[/imath]
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for [imath]\pi/4, \log 2[/imath] and similar alternating series etc. One series that popped into our discussio... | 329613 | How do I find [imath]\sum_{n=1}^\infty \frac{1}{n^n}[/imath]
I stumbled across this problem to find the result of the following expression: [imath]\sum_{n=1}^\infty \frac{1}{n^n}[/imath] but I don't know how to approach it. It was suggested to me that I try this: [imath]\sum_{n=1}^\infty e^{-n\ln n}[/imath] However, I... |
44113 | What's the value of [imath]\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}[/imath]?
For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. The question is motivated from the simple e... | 679790 | Value of [imath]\sum\limits_{n= 0}^\infty \frac{n²}{n!}[/imath]
How to compute the value of [imath]\sum\limits_{n= 0}^\infty \frac{n^2}{n!}[/imath] ? I started with the ratio test which told me that it converges but I don't know to what value it converges. I realized I only know how to calculate the limit of a power... |
17054 | Group where every element is order 2
Let [imath]G[/imath] be a group where every non-identity element has order 2. If |G| is finite then [imath]G[/imath] is isomorphic to the direct product [imath]\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \ldots \times \mathbb{Z}_{2}[/imath]. Is the analogous result [imath]G= ... | 1950783 | Let [imath]G[/imath] be a group. Prove that IF [imath]x^2 = e[/imath] for all [imath]x \in G[/imath], then [imath]G[/imath] is abelian.
Let [imath]G[/imath] be a group. Prove that IF [imath]x^2 = e[/imath] for all [imath]x \in G[/imath], then [imath]G[/imath] is abelian. My attempt: [imath]x^2 = e[/imath] [imath]x = x... |
95741 | Is there any difference between mapping and function?
I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from [imath]\mathbb R[/imath] to [imath]\mathbb R[/imath]. But I am not ok with this a... | 1720557 | What is the difference between linear mappings and linear functions?
Let [imath]V[/imath] and [imath]V'[/imath] be vector spaces over a field [imath]K[/imath]. A linear mapping [imath]f:V \to V'[/imath] is a mapping which preserves addition and scalar multiplication. My question is: what is the difference between line... |
10490 | Why is [imath]1^{\infty}[/imath] considered to be an indeterminate form
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by t... | 319764 | 1 to the power of infinity, why is it indeterminate?
I've been taught that [imath]1^\infty[/imath] is undetermined case. Why is it so? Isn't [imath]1*1*1...=1[/imath] whatever times you would multiply it? So if you take a limit, say [imath]\lim_{n\to\infty} 1^n[/imath], doesn't it converge to 1? So why would the limit... |
137277 | Constructing a subset not in [imath]\mathcal{B}(\mathbb{R})[/imath] explicitly
While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of [imath]\mathbb{R}[/imath] which we meet in everyday use is an element of Borel [imath]\sigma[/imath]-algebra [imath]\ma... | 1372120 | An example of Lebesgue measurable set but not Borel measurable besides the "subset of Cantor set" example.
The question is to give and example of Lebesgue measurable set but not Borel measurable. I know there exists subset of Cantor set that is not Borel measurable, since the cardinality of all Borel sets in [imath][0... |
107617 | An infinite subset of a countable set is countable
In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid? Let [imath]S[/imath] be an infinite subset of a... | 775464 | How to prove that this function is surjective?
I try to give a more constructive proof of the following lemma [imath]\qquad[/imath] Let [imath]S[/imath] be countably infinite and [imath]A[/imath] an infinite subset of [imath]S.[/imath] Then [imath]A[/imath] is countable. Here the "constructive proof" means that to pr... |
73348 | Can the Surface Area of a Sphere be found without using Integration?
When we were in school they told us that the Surface Area of a sphere = [imath]4\pi r^2[/imath] Now, when I try to derive it using only high school level mathematics, I am unable to do so. Please help. | 335577 | how to find surface area of a sphere
could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its [imath]4\pi r^2[/imath] if the sphere is of radius [imath]r[/imath], I also want to know what is the area of unit... |
30732 | How can I evaluate [imath]\sum_{n=0}^\infty(n+1)x^n[/imath]?
How can I evaluate [imath]\sum_{n=1}^\infty\frac{2n}{3^{n+1}}[/imath]? I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these be... | 301139 | Sum of a series using derivatives
[imath]1 + 2/2 + 3/4 + \cdots + n/2^{n-1}[/imath] How would find the closed-form expression and also the sum up to 20? I'm not really getting why or the logic behind using derivatives to arrive at an answer. |
303695 | if [imath]f[/imath] is differentiable at a point [imath]x[/imath], is [imath]f[/imath] also necessary lipshitz-continuous at [imath]x[/imath]?
if [imath]f[/imath] is differentiable at a point [imath]x[/imath], is [imath]f[/imath] also necessary Lipshitz at [imath]x[/imath]? Since [imath]f[/imath] is differentiable at ... | 286715 | Lipschitz condition: if [imath]f[/imath] is differentiable at [imath]b[/imath], then [imath]f[/imath] is Lipschitz of order [imath]1[/imath] at [imath]b[/imath]
I have been trying to solve this, but failing at it. Since [imath]f[/imath] is differentiable at [imath]x[/imath], we have [imath]f'(x)=\lim_{y \to x} \fra... |
11 | Is it true that [imath]0.999999999\dots=1[/imath]?
I'm told by smart people that [imath]0.999999999\dots=1[/imath] and I believe them, but is there a proof that explains why this is? | 419866 | Can someone help me solve this problem please.
For the real numbers [imath]x=0.9999999\dots[/imath] and [imath]y=1.0000000\dots[/imath] it is the case that [imath]x^2<y^2[/imath]. Is it true or false? Prove if you think it's true and give a counterexample if you think it's false. |
201906 | Showing that [imath]\frac{\sqrt[n]{n!}}{n}[/imath] [imath]\rightarrow \frac{1}{e}[/imath]
Show:[imath]\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}[/imath] So I can expand the numerator by geometric mean. Letting [imath]C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n[/imath]. Let the numerator be called [imath]a... | 935490 | The limit of [imath](n!)^{1/n}/n[/imath] as [imath]n\to\infty[/imath]
(Proof necessary) [imath]\lim_{n \to \infty} \frac{(n!)^{\frac{1}{n}}}{n}[/imath] I don't have an answer yet, but I know it exists, and is less than [imath]1[/imath]. Edit. Winther's answer is the most correct I don't understand how he is jumping ... |
189328 | Nonexistence of an injective [imath]C^1[/imath] map between [imath]\mathbb R^2[/imath] and [imath]\mathbb R[/imath]
I am getting bored waiting for the train so I'm thinking whether there can exist a [imath]C^1[/imath] injective map between [imath]\mathbb{R}^2[/imath] and [imath]\mathbb{R}[/imath]. It seems to me that ... | 1079075 | Is there a injective polynomial function from [imath]R^2[/imath] to [imath]R[/imath]?
There is an injective polynomial function from [imath]N^2[/imath] to [imath]N[/imath] (the Cantor-pairing function for example, which is of degree 2), and also one of degree 4 from [imath]Z^2[/imath] to [imath]Z[/imath]. I believe th... |
22069 | Is there a name for function with the exponential property [imath]f(x+y)=f(x) \cdot f(y)[/imath]?
I was wondering if there is a name for a function that satisfies the conditions [imath]f:\mathbb{R} \to \mathbb{R}[/imath] and [imath]f(x+y)=f(x) \cdot f(y)[/imath]? Thanks and regards! | 375801 | Is [imath]f(x)f(y)=f(x+y)[/imath] enough to determin [imath]f[/imath]?
I had a discussion with a friend and there it came up the question whether [imath]f(x)f(y)=f(x+y)[/imath], [imath]f(0)=1[/imath] and the existence of [imath]f'(x)[/imath] implies that [imath]f(x)=\exp(a x)[/imath]. This seems very reasonable but I ... |
170813 | Prove [imath](-a+b+c)(a-b+c)(a+b-c) \leq abc[/imath], where [imath]a, b[/imath] and [imath]c[/imath] are positive real numbers
I have tried the arithmetic-geometric inequality on [imath](-a+b+c)(a-b+c)(a+b-c)[/imath] which gives [imath](-a+b+c)(a-b+c)(a+b-c) \leq \left(\frac{a+b+c}{3}\right)^3[/imath] and on [imath]... | 2849480 | [imath]x[/imath], [imath]y[/imath] and [imath]z[/imath] are sides of a triangle - prove that [imath](x+y-z)(x-y+z)(-x+y+z)\leq xyz[/imath]
[imath]x[/imath], [imath]y[/imath] and [imath]z[/imath] are all sides of a triangle. Prove that [imath](x+y-z)(x-y+z)(-x+y+z)\leq xyz[/imath]. Equality occurs when all sides are th... |
13131 | Starting digits of [imath]2^n[/imath].
Prove that for any finite sequence of decimal digits, there exists an [imath]n[/imath] such that the decimal expansion of [imath]2^n[/imath] begins with these digits. | 544214 | Is [imath]2^k = 2013...[/imath] for some [imath]k[/imath]?
I'm wondering if some power of [imath]2[/imath] can be written in base [imath]10[/imath] as [imath]2013[/imath] followed by other digits. Formally, does there exist [imath]k,q,r \in \mathbb N[/imath] such that [imath]2^k=2013 \cdot 10^q+r \,\,\,; \,\,\,r<10^q ... |
296101 | An explicit bijection between the power set [imath]\mathcal P \left({\mathbb{N}}\right)[/imath] and [imath]2^\mathbb{N}[/imath].
I know how to show that these two have the same cardinality and from that there must be a bijection between them. Can anyone help with an explicit bijection between these sets? | 41006 | How to show equinumerosity of the powerset of [imath]A[/imath] and the set of functions from [imath]A[/imath] to [imath]\{0,1\}[/imath] without cardinal arithmetic?
How to show equinumerosity of the powerset of [imath]A[/imath] and the set of functions from [imath]A[/imath] to [imath]\{0,1\}[/imath] without cardinal... |
4467 | How to prove: if [imath]a,b \in \mathbb N[/imath], then [imath]a^{1/b}[/imath] is an integer or an irrational number?
It is well known that [imath]\sqrt{2}[/imath] is irrational, and by modifying the proof (replacing 'even' with 'divisible by [imath]3[/imath]'), one can prove that [imath]\sqrt{3}[/imath] is irrational... | 449974 | Prove [imath]\sqrt{k}[/imath] is not a rational number.
Suppose [imath]k>1[/imath] is an integer, and k is not a square number, then [imath]\sqrt{k}[/imath] is not a rational number. Proof: Let [imath]\sqrt{k}=\frac{p}{q}[/imath], and [imath](p,q)=1[/imath],So [imath]q^2|p^2[/imath], [imath]p\neq 1[/imath], [imath]k[/... |
29450 | Self-Contained Proof that [imath]\sum\limits_{n=1}^{\infty} \frac1{n^p}[/imath] Converges for [imath]p > 1[/imath]
To prove the convergence of the p-series [imath]\sum_{n=1}^{\infty} \frac1{n^p}[/imath] for [imath]p > 1[/imath], one typically appeals to either the Integral Test or the Cauchy Condensation Test. I am w... | 400604 | Prove the convergence of the series.
Let r > 1 be a real number. Prove that the following series is convergent. [imath]\sum_{n = 1}^{\infty}\frac{1}{n^r}[/imath] |
48161 | In classical logic, why is [imath](p\Rightarrow q)[/imath] True if both [imath]p[/imath] and [imath]q[/imath] are False?
I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement [imath](p \Rightarrow q)\;[/imath] (a.k.a. '[imath]p[/imath] implies [imath]q[/... | 636224 | Why is the implication [imath]P \Rightarrow Q[/imath] false if and only if [imath]P[/imath] is true and [imath]Q[/imath] is false?
Why is the implication [imath]P \Rightarrow Q[/imath] false if and only if [imath]P[/imath] is true and [imath]Q[/imath] is false ? Is this because if [imath]P[/imath] implies [imath]Q[/im... |
136665 | For every irrational [imath]\alpha[/imath], the set [imath]\{a+b\alpha: a,b\in \mathbb{Z}\}[/imath] is dense in [imath]\mathbb R[/imath]
I am not able to prove that this set is dense in [imath]\mathbb{R}[/imath]. Will be pleased if you help in a easiest way, [imath]\{a+b\alpha: a,b\in \mathbb{Z}\}[/imath] where [imath... | 1437468 | Denseness of the set [imath]\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}[/imath] with [imath]\alpha[/imath] irrational
How to prove that the set [imath]\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}[/imath], ([imath]\alpha[/imath] is an irrational number) is dense in [imath]\mathbb{R}[/imath]? Using the fact every... |
125709 | Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
Let [imath]A[/imath] and [imath]B[/imath] be two sets of real numbers. Define the distance from [imath]A[/imath] to [imath]B[/imath] by [imath]\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.[/imath] Give an example to s... | 539687 | I don't understand how sets can be closed, yet disjoint?
What are some closed, disjoint subsets [imath]A, B[/imath] in [imath]R^2[/imath] where [imath]inf\{d(A, B) = 0 \forall a \in A \forall b \in B\}[/imath]? |
72856 | Order of finite fields is [imath]p^n[/imath]
Let [imath]F[/imath] be a finite field. .How do I prove that the order of [imath]F[/imath] is always of order [imath]p^n[/imath] where [imath]p[/imath] is prime? | 317633 | Is there a field with [imath]n[/imath] elements for all [imath]n \in \mathbb{N}[/imath]?
I don't think this is true, but I'm not sure. I certainly know of finite fields with 2,4 and 8 elements, and of course [imath]p^n[/imath] elements where [imath]p[/imath] is prime, for all [imath]n \in \mathbb{N}[/imath]. |
168812 | A compact operator is completely continuous.
I have a question. If [imath]X[/imath] and [imath]Y[/imath] are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping [imath]T \colon X \to Y[/imath] is called completely continuous, if it maps a weakly convergent sequence in [i... | 811219 | Problems proving that a compact operator is completely continuous
I would like to prove that if [imath]T:X\rightarrow Y[/imath] is a compact operator, then for every weak convergent sequence [imath](x_n)_{n\in\mathbb N}[/imath] with [imath]x_n\rightharpoonup x[/imath] for some [imath]x\in X[/imath] it follows that [im... |
236578 | Recurrence relation, Fibonacci numbers
[imath](a)[/imath] Consider the recurrence relation [imath]a_{n+2}a_n = a^2 _{n+1} + 2[/imath] with [imath]a_1 = a_2 = 1[/imath]. [imath](i)[/imath] Assume that all [imath]a_n[/imath] are integers. Prove that they are all odd and the integers [imath]a_n[/imath] and [imath]a_{n+1}... | 241663 | Recurrence relation, induction and Fibonacci numbers
1.(a) Consider the recurrence relation [imath]a_{n+2}a_n = a^2_{n+1} + 2[/imath] with [imath]a_1 = a_2 = 1[/imath]. (i) Assume that all [imath]a_n[/imath] are integers. Prove that they are all odd and the integers [imath]a_n[/imath] and [imath]a_{n+1}[/imath] are co... |
278330 | Consider the series [imath] ∑_{n=1}^∞ x^2+ n/n^2[/imath] . Pick out the true statements:
Consider the series [imath] \sum_{n=1}^\infty x^2+ n/n^2[/imath] . Pick out the true statements: (a) The series converges for all real values of [imath]x[/imath]. (b) The series converges uniformly on [imath]\mathbb{R}[/imath]. (... | 641336 | different convergence of the series [imath]\sum_1^\infty{(-1)^n\frac{x^2+n}{n^2}}[/imath]
Consider the series [imath]\sum_1^\infty{(-1)^n\frac{x^2+n}{n^2}}[/imath] Pick out the true statements: (a) The series converges for all real values of [imath]x[/imath]. (b) The series converges uniformly on [imath]\mathbb{R}[/im... |
8337 | Different methods to compute [imath]\sum\limits_{k=1}^\infty \frac{1}{k^2}[/imath] (Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) [imath]\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.[/imath] However, Euler was Euler and he gav... | 302310 | Show that [imath]\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}[/imath]
Show that [imath]\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}[/imath] Anyone can help ? |
207395 | Limit of a continuous function
Suppose that [imath]f[/imath] is a continuous and real function on [imath][0,\infty][/imath]. How can we show that if [imath]\lim_{n\rightarrow\infty}(f(na))=0[/imath] for all [imath]a>0[/imath] then [imath]\lim_{x\rightarrow+\infty} f(x)=0[/imath]? | 1935620 | Example: Continuous function having a limit along every arithmetic sequence but having no limit over the reals
The following question seems to be very elementary and must be a folklore, but we are not able to find an answer. Let [imath]f: [0,\infty)\to \mathbb R[/imath] be a continuous function such that for every [im... |
3852 | If [imath]AB = I[/imath] then [imath]BA = I[/imath]
If [imath]A[/imath] and [imath]B[/imath] are square matrices such that [imath]AB = I[/imath], where [imath]I[/imath] is the identity matrix, show that [imath]BA = I[/imath]. I do not understand anything more than the following. Elementary row operations. Linear d... | 462983 | Prove that a square matrix commutes with its inverse
The Question: This is a very fundamental and commonly used result in linear algebra, but I haven't been able to find a proof or prove it myself. The statement is as follows: let [imath]A[/imath] be an [imath]n\times n[/imath] square matrix, and suppose that [imath... |
28438 | Alternative proof that [imath](a^2+b^2)/(ab+1)[/imath] is a square when it's an integer
Let [imath]a,b[/imath] be positive integers. When [imath]k = \frac{a^2 + b^2}{ab+1}[/imath] is an integer, it is a square. Proof 1: (Ngô Bảo Châu): Rearrange to get [imath]a^2-akb+b^2-k=0[/imath], as a quadratic in [imath]a[/imath... | 372200 | To prove [imath]\frac {a^2+b^2}{ab+1}[/imath] is a perfect square , without geometry or induction.
Let [imath]a[/imath] and [imath]b[/imath] be positive integers such that [imath]ab+1[/imath] divides [imath]a^2+b^2[/imath] ; then prove that [imath]\frac {a^2+b^2}{ab+1}[/imath] is a perfect square (this problem came in... |
54210 | Is [imath]x^x=y[/imath] solvable for [imath]x[/imath]?
Given that [imath]x^x = y[/imath]; and given some value for [imath]y[/imath] is there a way to expressly solve that equation for [imath]x[/imath]? | 351046 | Solve [imath]x^x = a[/imath] for known [imath]a[/imath]?
For example if you have [imath]x^x = 2[/imath], can you express [imath]x[/imath] as a numerical expression containing only the addition, multiplication and exponentiation operators? |
300904 | Prove that G and G' are isomorphic.
Let [imath]G=[/imath] the set of [imath]2\times 2[/imath] matrices [imath]\left\{\begin{bmatrix} a& b\\ 0& a-2b\end{bmatrix}\;\middle\vert\; a,b \in\mathbb{R} \text{ and }A^2\neq 2ab\right\},[/imath] where the group operation is matrix multiplication. Let [imath]G'=\{(c,d) \mid c,... | 300338 | Need help is defining an isomorphism.
Let [imath]A = \left\{ \begin{bmatrix} a & b \\ 0 & {a-2b} \end{bmatrix} \mid a,b \in \mathbb{R}, a^2 \ne 2ab \right\}[/imath] where the group operation is matrix multiplication. Let [imath]A'= \left\{(c,d) \mid c,d\in \mathbb{R}, c,d \ne 0\right\}[/imath] with group operation [... |
14721 | Three variable, third degree Diophantine equation
I haven't found any useful method to solve the following problem: Prove that if [imath]x,y,z\in\mathbb{Z}[/imath] and [imath]x^3+y^3=3z^3[/imath] then [imath]xyz=0[/imath]. Source: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=382377 | 473610 | a diophantine equation from Stewart and Tall
This is from Stewart and Tall from the chapter on Kummer's Theorem. Show that there are no non trivial (non-zero) solutions to [imath]x^3 + y^3=3z^3[/imath] |
15591 | Number of even and odd subsets
Suppose we have the following two identities: [imath]\displaystyle \sum_{k=0}^{n} \binom{n}{k} = 2^n[/imath] [imath]\displaystyle \sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0[/imath] The first says that the number of subsets of an [imath]n[/imath]-set is [imath]2^n[/imath]. The second say... | 94514 | Proving [imath]\sum_{k=0}^n(-1)^k\binom nk=0[/imath]
Show that [imath]\sum_{k=0}^n(-1)^k\binom nk=0[/imath] So for odd [imath]n[/imath] we have an even number of terms. So [imath]\binom nk=\binom n{n-k}[/imath] which have opposite signs. Thus the sum is 0. For even [imath]n[/imath] we have that [imath]\sum_{k=0}^n(-1)... |
16374 | Universal Chord Theorem
Let [imath]f \in C[0,1][/imath] and [imath]f(0)=f(1)[/imath]. How do we prove [imath]\exists a \in [0,1/2][/imath] such that [imath]f(a)=f(a+1/2)[/imath]? In fact, for every positive integer [imath]n[/imath], there is some [imath]a[/imath], such that [imath]f(a) = f(a+\frac{1}{n})[/imath]. For... | 371571 | Prove that there exists [imath]x_n[/imath] such that [imath]0 \leq x_n \leq 1-\frac{1}{n}[/imath] and [imath]f(x_n)=f(x_n+\frac{1}{n})[/imath].
Suppose that the function [imath]f:[0,1] \to \mathbb{R}[/imath] is continuous on [imath][0,1][/imath] and [imath]f(0)=f(1)[/imath]. Prove that for each natural number [imat... |
61087 | How would one go about proving that the rationals are not the countable intersection of open sets?
I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why [imath]\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q... | 2765171 | Set of rational number in (0,1) can not be expressed as intersection of countable collection of open sets
Set S=Q[imath]\cap[/imath](0,1) I wanted to show that this can not be expressed as intersection of countable collection of open sets. I know Any closed set can be shown as intersection of countable collection of o... |
151864 | Continuous images of compact sets are compact
Let [imath]X[/imath] be a compact metric space and [imath]Y[/imath] any metric space. If [imath]f:X \to Y[/imath] is continuous, then [imath]f(X)[/imath] is compact (that is, continuous functions carry compact sets into compact sets). Proof: Consider an open cover of [im... | 2047150 | If [imath]h:X \to Y[/imath] is a homeomorphism of metric spaces, how do I prove that [imath]X[/imath] is compact if and only if [imath]Y[/imath] is compact?
If [imath]h:X \to Y[/imath] is a homeomorphism of metric spaces, how do I prove that [imath]X[/imath] is compact if and only if [imath]Y[/imath] is compact? I c... |
139659 | Total number of solutions of an equation
What is the total number of solutions of an equation of the form [imath]x_1 + x_2 + \cdots + x_r = m[/imath] such that [imath]1 \le x_1 < x_2 < \cdots < x_r < N[/imath] where [imath]N[/imath] is some natural number and [imath]x_1, x_2, \cdots, x_r, m[/imath] are integers? Also,... | 1653130 | How many solutions are [imath] a+b+c+d = 30 ,(a\leq b\leq c\leq d) [/imath]?
I would appreciate if somebody could help me with the following problem: Q: How many solutions are there to the equation [imath] a+b+c+d = 30 ,(a\leq b\leq c\leq d) [/imath] where [imath]a,b,c,d\in \{1,2,\cdots,30\}[/imath] |
15423 | Optimal algorithm for finding the odd sphere with a balance scale
Say we have [imath]N[/imath] spheres indexed as [imath]1,2,3,\dotsc, N[/imath] such that all of them have identical weight apart from one, and we don't know if that one is heavier or lighter. We have to determine which sphere has the odd weight using ju... | 2711627 | Balls and scales (generalization)
A classical brain teaser is : You have twelve balls identical in size and appearance but one ball is an odd weight (could be either light or heavy). You have a set of balance scales. How many steps are necessary to identify the odd ball and to say whether it is light or heavy? The a... |
26722 | calculating [imath]a^b \!\mod c[/imath]
What is the fastest way (general method) to calculate the quantity [imath]a^b \!\mod c[/imath]? For example [imath]a=2205[/imath], [imath]b=23[/imath], [imath]c=4891[/imath]. | 81228 | How do I compute [imath]a^b\,\bmod c[/imath] by hand?
How do I efficiently compute [imath]a^b\,\bmod c[/imath]: When [imath]b[/imath] is huge, for instance [imath]5^{844325}\,\bmod 21[/imath]? When [imath]b[/imath] is less than [imath]c[/imath] but it would still be a lot of work to multiply [imath]a[/imath] by itsel... |
254865 | Simple binomial theorem proof: [imath]\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k[/imath]
I am trying to prove this binomial statement: For [imath]a \in \mathbb{C}[/imath] and [imath]k \in \mathbb{N_0}[/imath], [imath]\sum_{j=0}^{k} {a+j \choose j} = {a+k+1 \choose k}.[/imath] I am stuck where and how to start. My... | 866046 | Identity with binomials
Does there exist a closed formula for [imath]\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?[/imath] I've searching on wikipedia but I haven't found this kind of sum. |
91087 | Subfields of finite fields
We know that if a finite field [imath]F[/imath] has characteristic [imath]p[/imath] (prime), then [imath]F[/imath] has cardinality [imath]p^r[/imath] where [imath]r = [F:\mathbb{F}_p][/imath]. I'm now trying to say something about the possible cardinalities of subfields of [imath]F[/imath].... | 1377867 | I have to show no proper intermediate fields exist between [imath]Z_2[/imath] and [imath]GF(2^3)[/imath]
I have to show no proper intermediate fields exist between [imath]Z_2[/imath] and its overfield [imath]GF(2^3)[/imath], Can any one help? |
96739 | Do continuous linear functions between Banach spaces extend?
Just wondering... Let [imath]E[/imath], [imath]G[/imath] be Banach spaces, let [imath]U\subset E[/imath] be a subset of [imath]E[/imath], and let [imath]f:U\rightarrow G[/imath] be a continuous linear function. Can [imath]f[/imath] be extended to a continuou... | 1182253 | Reference request: linear operators into [imath]L^\infty[/imath] can be extended presrving the norm.
Suppose [imath]X[/imath] is a normed linear space and [imath]Y\subset X[/imath] a linear subspace. I remember that any linear map [imath]L\colon Y\to L^\infty(\Omega)[/imath] can be extended to a linear map [imath]\til... |
296745 | There exist a function such that [imath]f\circ f(x)=e^x[/imath]?
Based on this question: How to calculate [imath]f(x)[/imath] in [imath]f(f(x)) = e^x[/imath]? I would like to know if I can get a function such that [imath]f:\mathbb R \to \mathbb R^+[/imath], defined by [imath]f\circ f(x)=e^x[/imath]. My guess is no, b... | 59023 | How to calculate [imath]f(x)[/imath] in [imath]f(f(x)) = e^x[/imath]?
How would I calculate the power series of [imath]f(x)[/imath] if [imath]f(f(x)) = e^x[/imath]? Is there a faster-converging method than power series for fractional iteration/functional square roots? |
15129 | how can be prove that [imath]\max(f(n),g(n)) = \Theta(f(n)+g(n))[/imath]
how can be prove that [imath]\max(f(n),g(n)) = \Theta(f(n)+g(n))[/imath] though the big O case is simple since [imath]\max(f(n),g(n)) \leq f(n)+g(n)[/imath] edit : where [imath]f(n)[/imath] and [imath]g(n)[/imath] are asymptotically nonnegative f... | 267252 | How to prove that [imath]\max(f(n), g(n)) = \Theta(f(n) + g(n))[/imath]?
Using the basic definition of theta notation prove that [imath]\max(f(n), g(n)) = \Theta(f(n) + g(n))[/imath] I came across two answer to this question on this website but the answers weren't clear to me. Would you mind to elaborate how this ca... |
102476 | [imath]f\geq 0[/imath], continuous and [imath]\int_a^b f=0[/imath] implies [imath]f=0[/imath] everywhere on [imath][a,b][/imath]
This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis. Problem 6.2: Suppose [imath]f\geq 0[/imath], f is continuous on [imath][a, b][/imath], and [imath]\int_a^b ... | 1750372 | Show that [imath]f[/imath] is identically zero if and only if [imath]\int_a^b f(x)dx = 0[/imath]
Assume [imath]f[/imath] is continuous and nonnegative over [imath][a,b][/imath]. Show that [imath]f[/imath] is identically zero if and only if [imath]\displaystyle \int_{a}^b f(x)dx = 0[/imath]. Proving the first directi... |
197393 | Why does [imath]\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi[/imath]?
Playing around on wolframalpha shows [imath]\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi[/imath]. I know [imath]\tan^{-1}(1)=\pi/4[/imath], but how could you compute that [imath]\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi[/imath] to get this result? | 1657856 | proofs on trigonometric identities involving complex numbers
Provide a reason for each step of the proof. Prove the identity [imath]\arctan(1)+\arctan(2)+\arctan(3)=180^{\circ}[/imath]Proof:\begin{align*}\arctan(1)+\arctan(2)+\arctan(3)&=\text{arg}(1+i)+\text{arg}(1+2i)+\text{arg}(1+3i)\\&=\text{arg}\left[(1+i)(1+2i)(... |
18983 | why is [imath]\sum\limits_{k=1}^{n} k^m[/imath] a polynomial with degree [imath]m+1[/imath] in [imath]n[/imath]
why is [imath]\sum\limits_{k=1}^{n} k^m[/imath] a polynomial with degree [imath]m+1[/imath] in [imath]n[/imath]? I know this is well-known. But how to prove it rigorously? Even mathematical induction does n... | 731047 | Sum of series: [imath]1^k+2^k+3^k+...+n^k =?[/imath]
Is there any better algorithm to solve this equation other than brute force. [imath]1^k+2^k+3^k+...+n^k=[/imath] formula? Here [imath]k[/imath] is a natural number, [imath]n[/imath] is a natural number. |
84392 | Why do the [imath]n \times n[/imath] non-singular matrices form an "open" set?
Why is the set of [imath]n\times n[/imath] real, non-singular matrices an open subset of the set of all [imath]n\times n[/imath] real matrices? I don't quite understand what "open" means in this context. Thank you. | 430488 | Topology of Lie groups.
I do not understand the topology of a Lie group clearly. Let [imath]G[/imath] be a Lie group and [imath]T_eG[/imath] be its tangent space at the identity [imath]e \in G[/imath]. Why [imath]Aut(T_eG)[/imath] is an open subset of the vector space of endomorphisms of [imath]T_eG[/imath] (i.e. [ima... |
298614 | General term of [imath]a_n = 2a_{n-1} + 1[/imath]
Find the general term of the sequence defined by: [imath]a_n = 2a_{n-1} + 1[/imath] where [imath]a_1[/imath] is given Thank You | 183599 | can one derive the [imath]n^{th}[/imath] term for the series, [imath]u_{n+1}=2u_{n}+1[/imath],[imath]u_{0}=0[/imath], [imath]n[/imath] is a non-negative integer
derive the [imath]n^{th}[/imath] term for the series [imath]0,1,3,7,15,31,63,127,255,\ldots[/imath] observation gives, [imath]t_{n}=2^n-1[/imath], where [imat... |
78560 | How do you solve the Initial value probelm [imath]dp/dt = 10p(1-p), p(0)=0.1[/imath]?
The problem is... [imath] \frac{dp}{dt} = 10p(1-p),[/imath] [imath]p(0)=0.1[/imath]. Solve and show that [imath]p(t) \to 1[/imath] as [imath]t\to \infty.[/imath] I know this is probably really simple, I was trying to go down the l... | 1070953 | Differential equation problem. Integrating the logistic equation.
I would like to know how to integrate or rather solve this: [imath] \frac{dP}{dt} = kP(L-P). [/imath] I have the solution, but I would like to know how to arrive at it. I have been told it involves separation of variables and partial fractions. |
61828 | Proof of Frullani's theorem
How can I prove the Theorem of Frullani? I did not even know all the hypothesis that [imath]f[/imath] must satisfy, but I think that this are Let [imath]\,f:\left[ {0,\infty } \right) \to \mathbb R[/imath] be a a continuously differentiable function such that [imath] \mathop {\lim }\limits_... | 1046074 | Show that [imath]\int_0^{\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-L]\ln\frac{b}{a}[/imath]
Let [imath]f:[0,\infty)\to\mathbb{R}[/imath] be continuous and [imath]\lim_{x\to\infty}f(x)=L[/imath]. Show that [imath]\int_0^{\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-L]\ln\frac{b}{a}[/imath] where [imath]0<a<b[/imath]. I don't even k... |
9286 | Proving [imath]\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}[/imath]
How to prove [imath]\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}[/imath] | 355024 | How to calculate the following integral?
How to calculate the following integral? [imath]\int_{- \infty}^{\infty} \mathrm{e}^{- \frac{x^2}{2}} \mathrm{d} x[/imath] |
187729 | Evaluating [imath]\int_0^\infty \sin x^2\, dx[/imath] with real methods?
I have seen the Fresnel integral [imath]\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}[/imath] evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integra... | 1142514 | Evaluate [imath]\int_{0}^{\infty} \cos(x^2)dx [/imath]
Prove that the above integral is equal to [imath]\frac{\sqrt{2\pi}}{2}[/imath] I have already tried expanding using [imath]\cos[/imath] identity and also taking Laplace for it. I am getting nowhere with this. |
14666 | Number of permutations of [imath]n[/imath] elements where no number [imath]i[/imath] is in position [imath]i[/imath]
I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, [imath]3,1,5,2,4[/imath] is an acceptable permutation where [im... | 408341 | A basic probability doubt on derangment
Is there any implication that the probability that a random permutation is a derangment is [imath]\frac{1}{e}[/imath] when [imath]n->\infty[/imath] ? |
95799 | Why [imath]\gcd(qb+r,b)=\gcd(b,r)[/imath]?
Given: [imath]a = qb + r[/imath]. Then it holds that [imath]\gcd(a,b)=\gcd(b,r)[/imath]. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math.stackexchange.com/a/4110/53259 and averting a duplicate)... | 488661 | If [imath]a = r \pmod b[/imath] when [imath]0 \le r < b[/imath], then [imath]\gcd(a, b) = \gcd (b, r)[/imath]
If [imath]a = r \pmod b[/imath] when [imath]0 \le r < b[/imath], then [imath]\gcd(a, b) = \gcd (b, r)[/imath]. I do not understand why this is, can somebody explain? I've also looked over this thread: Why is ... |
51502 | If [imath]f_k \to f[/imath] a.e. and the [imath]L^p[/imath] norms converge, then [imath]f_k \to f[/imath] in [imath]L^p[/imath]
Let [imath]1\leq p < \infty[/imath]. Suppose that [imath]\{f_k\} \subset L^p[/imath] (the domain here does not necessarily have to be finite), [imath]f_k \to f[/imath] almost everywhere, a... | 592586 | Basic [imath]L_1[/imath] convergence question
I think this is quite a simple question, but for some reason am finding it difficult to answer. The question is: If [imath](f_n:n\in\mathbb{N})[/imath] is a sequence of integrable functions, with [imath]f_n \to f[/imath] a.e. for some integrable [imath]f[/imath], then is i... |
18690 | Algebraic Proof that [imath]\sum\limits_{i=0}^n \binom{n}{i}=2^n[/imath]
I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of [imath]n[/imath] elements. I'm curious if there's a series of algebraic manipulations that... | 59554 | Algebraic proof that collection of all subsets of a set (power set) of [imath]N[/imath] elements has [imath]2^N[/imath] elements
In other words, is there an algebraic proof showing that [imath]\sum_{k=0}^{N} {N\choose k} = 2^N[/imath]? I've been trying to do it some some time now, but I can't seem to figure it out. |
63870 | A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem
When I google "baire category theorem", I get a link to Ben Green's website. And at the end of the paper, he mentioned such a classic problem: Suppose that [imath]f:\mathbb{R}^+\to\mathbb{R}^+[/imath] is ... | 101086 | [imath]\lim_{n\to \infty}f(nx)=0[/imath] implies [imath]\lim_{x\to \infty}f(x)=0[/imath]
Can anyone help me with this problem? Let [imath]f:[0,\infty)\longrightarrow \mathbb R[/imath] be a continuous function such that for each [imath]x>0[/imath], we have [imath]\lim_{n\to \infty}f(nx)=0[/imath]. Then prove that [ima... |
24060 | Uniform continuity on (0,1) implies boundedness
I need to prove that if [imath]f: (0,1) \rightarrow \mathbb{R}[/imath] is Uniformly continuous then it is bounded. Thank you. | 567577 | Why does uniform continuity of a function imply that the function is bounded?
As the title states, I'm wondering why: If [imath]A[/imath] is a bounded subset of [imath]\mathbb{R}[/imath] and [imath]f:A\to \mathbb{R}[/imath] is uniformly continuous on [imath]A[/imath], then [imath]f[/imath] must be bounded on [imath]A[... |
67620 | Set of continuity points of a real function
I have a question about subsets [imath] A \subseteq \mathbb R [/imath] for which there exists a function [imath]f : \mathbb R \to \mathbb R[/imath] such that the set of continuity points of [imath]f[/imath] is [imath]A[/imath]. Can I characterize this kind of sets? In a to... | 64677 | characteristic function of the rationals
Let [imath]\chi[/imath] be the characteristic function of the rational numbers in [imath][0,1][/imath]. Does there exist a sequence [imath]\{f_n\}[/imath] of continuous functions on [imath][0,1][/imath] that converges pointwise to [imath]\chi[/imath]? |
141774 | Choice of [imath]q[/imath] in Baby Rudin's Example 1.1
First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2. In the second version of the proof, sho... | 511228 | Intuition in Rudin's Proof on Page 2
Rudin has a proof in which he is proving that [imath]A=\{ p \in \mathbb{Q}\;|\; p^2<2\}[/imath] has no maximum element (or in other words, an element which is greater than every other element). For this he creates a rational [imath]q=p-\dfrac{p^2-2}{p+2}[/imath] and uses this ratio... |
199026 | Don't we need the axiom of choice to choose from a non-empty set?
I recently read a proof that had the following in it: "since [imath]A[/imath] is non-empty, we can find an element [imath]x[/imath] in [imath]A[/imath]." This proof did not mention the axiom of choice, but it seems to me that it would be required to mak... | 2786279 | Schröder–Bernstein theorem for finite sets
I want to prove Schröder–Bernstein theorem for 2 finite sets [imath]A, B[/imath] of the same cardinal. I do it with induction on the cardinal number of [imath]A, B[/imath]. In the inducative step, I write: Let [imath]A, B[/imath] sets such that [imath]|A|=|B|=n+1[/imath] for... |
668 | What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a [imath]2\times... | 580754 | Physical meaning of the determinant .
In my undergraduate mechanics class, we just finished the section on coupled oscillators where we use eigenvalues to find solutions to the differential equations describing the motion of our spring mass system. When we finished, one of my fellow students started going on a rant ab... |
13801 | What's the thing with [imath]\sqrt{-1} = i[/imath]
What's the thing with [imath]\sqrt{-1} = i[/imath]? Do they really teach this in the US? It makes very little sense, because [imath]-i[/imath] is also a square root of [imath]-1[/imath], and the choice of which root to label as [imath]i[/imath] is arbitrary. So saying... | 1097134 | why is [imath]\sqrt{-1} = i[/imath] and not [imath]\pm i[/imath]?
this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE. Why is [imath]\sqrt{-1} = i[/imath] ... |
2260 | Proof for formula for sum of sequence [imath]1+2+3+\ldots+n[/imath]?
Apparently [imath]1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2[/imath]. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is known as "The sum of the first [imath]n[/imath] positive integers". | 658705 | What is the sum of integers from [imath]1[/imath] to [imath]789999[/imath] ? asks the professor
How to resolve it? How to find that sum? |
264980 | How to prove that [imath] \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | [/imath] diverges?
It is stated as a problem in Spivak's Calculus and I can't wrap my head around it. | 1897860 | How to prove that [imath]\sum_{n=1}^{\infty} \frac{|\sin n|}{n}[/imath] and [imath]\sum_{n=1}^{\infty} \frac{\sin^2 n}{n}[/imath] both diverge?
How can I prove that [imath]\sum_{n=1}^{\infty} \frac{|\sin n|}{n}[/imath] and [imath]\sum_{n=1}^{\infty} \frac{\sin^2 n}{n}[/imath] both diverge? I thought of using Compariso... |
92105 | [imath]f[/imath] uniformly continuous and [imath]\int_a^\infty f(x)\,dx[/imath] converges imply [imath]\lim_{x \to \infty} f(x) = 0[/imath]
Trying to solve [imath]f(x)[/imath] is uniformly continuous in the range of [imath][0, +\infty)[/imath] and [imath]\int_a^\infty f(x)dx [/imath] converges. I need to prove that... | 397214 | uniformly continuous and [imath]\int_0^\infty f(t)\,\mathrm dt[/imath] exists [imath]\implies \lim_{x\to\infty}f(x) = 0 [/imath]
I appreciate your help with this one. Let [imath]f \colon[0,\infty)\rightarrow \mathbb{R}[/imath] be uniformly continuous and let the integral [imath]\int_0^\infty f(t)\,\mathrm dt[/imath] e... |
305023 | Is the graph of every real function a null set?
This question popped to my mind during an analysis lecture: Let [imath]f:\mathbb{R} \rightarrow \mathbb{R}[/imath] be a (general) function. Is there an [imath]N\subset \mathbb{R}^2[/imath] with [imath]\lambda^2(N)=0[/imath], such that [imath]\{(x,f(x)):x\in \mathbb{R}\}[... | 35606 | Lebesgue Measure of the Graph of a Function
Let [imath]f:R^n \rightarrow R^m[/imath] be any function. Will the graph of [imath]f[/imath] always have Lebesgue measure zero? [imath](1)[/imath] I could prove that this is true if [imath]f[/imath] is continuous. [imath](2)[/imath] I suspect it is true if [imath]f[/imath] i... |
4551 | How can I prove [imath]\sup(A+B)=\sup A+\sup B[/imath] if [imath]A+B=\{a+b\mid a\in A, b\in B\}[/imath]
If [imath]A,B[/imath] non empty, upper bounded sets and [imath]A+B=\{a+b\mid a\in A, b\in B\}[/imath], how can I prove that [imath]\sup(A+B)=\sup A+\sup B[/imath]? | 509661 | Taking suprema on a set-equality
Suppose we have the following: [imath]X = A + B [/imath] where [imath]X, A,[/imath] and [imath]B[/imath] are any sets. [imath]A + B = \{ a + b : a \in A , \; \; \; b \in B \} [/imath] Can we conclude that [imath]\sup X = \sup A + \sup B [/imath] ? |
13490 | Proving that the sequence [imath]F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}[/imath] is boundedly convergent on [imath]\mathbb{R}[/imath]
Here is an exercise, on analysis which i am stuck. How do I prove that if [imath]F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}[/imath], then the sequence [imath]\{F_{n}(... | 1528099 | Problem dealing with [imath]\sum \frac{\sin(n)}{n}[/imath] and its convergence
[imath]\text{If} \ S=\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin (n)}{n}, \ \text{then what is} \ 2S+1[/imath] I know that [imath]\sum \frac{\sin(n)}{n}[/imath] converges. But now what do I do? |
21792 | Norms Induced by Inner Products and the Parallelogram Law
Let [imath] V [/imath] be a normed vector space (over [imath]\mathbb{R}[/imath], say, for simplicity) with norm [imath] \lVert\cdot\rVert[/imath]. It's not hard to show that if [imath]\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}[/imath] for some (r... | 370697 | Parallelogram law, dot product
Prove that if [imath]||\cdot||[/imath] satisfies [imath]||u-v||^2 + ||u+v||^2 = 2(||u||^2 + ||v||^2)[/imath] , then [imath]u \cdot v = \frac{1}{2} (||u+v||^2 - ||u||^2 - ||v||^2)[/imath] is dot product and [imath]||u||^2 = u \cdot u[/imath]. I've already shown that [imath](u+w)\cdot v = ... |
29023 | Value of [imath]\sum\limits_n x^n[/imath]
Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3 ? \end{equation*} Can we generalize the above to [imath]\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}[/imath] ? Are there some values of [imath]x[/im... | 68572 | Show that the geometric series [imath]a + ar +ar^2 + \cdots + ar^{n-1} + \cdots[/imath] converges if and only if [imath]|r| < 1[/imath]
Show, rigorously, that the geometric series [imath]a + ar +ar^2 + \cdots + ar^{n-1} + \cdots[/imath] converges if and only if |r| < 1. Also, show that if |r| < 1, the sum is given by... |
6979 | Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of [imath]y = f(x)[/imath] about the [imath]x[/imath]-axis from [imath]a[/imath] to [imath]b[/imath]. Then (using the disk method) the volume is [imath]\int_a^b \pi f(x)^2 dx[/imath] since we approximate... | 325547 | Why does the surface area integral need the arc length differential but the volume doesn't?
When calculating the surface area of a revolution you need to use the arc length differential [imath]\sqrt{1 + y'^2}[/imath] but you don't need to use that when calculating the volume. Why is that? Thanks! |
29366 | Do sets, whose power sets have the same cardinality, have the same cardinality?
Is it generally true that if [imath]|P(A)|=|P(B)|[/imath] then [imath]|A|=|B|[/imath]? Why? Thanks. | 842789 | How do we prove that, if [imath]\mathcal{P}(A) \sim \mathcal{P}(B)[/imath], then [imath]A \sim B[/imath]?
The converse--if [imath]\ A \sim B[/imath] then [imath] \mathcal{P}(A) \sim \mathcal{P}(B)[/imath]--is very easy to prove. I can't see an immediate, simple proof for the converse case. It seems like a potentially ... |
191453 | How to show that the modulus of [imath]\frac{z-w}{1-\bar{z}w}[/imath] is always [imath]1[/imath]?
Let's suppose that [imath]|z|<1[/imath] and [imath]|w|=1[/imath]. Show that the modulus of [imath]\displaystyle \frac{z-w}{1-\bar{z}w}[/imath] is always [imath]1[/imath]. Some hint. | 1613426 | Prove that [imath]|\frac{a-b}{1-\bar ab}|=1[/imath] if [imath]|a|=1[/imath] or [imath]|b|=1[/imath]
Prove that [imath]|\frac{a-b}{1-\bar ab}|=1[/imath] if [imath]|a|=1[/imath] or [imath]|b|=1[/imath] I assumed [imath]|a|=1[/imath]. Then tried to show that our statement holds. I wrote [imath]a=a_1+ia_2[/imath] and [i... |
6244 | Is there a quick proof as to why the vector space of [imath]\mathbb{R}[/imath] over [imath]\mathbb{Q}[/imath] is infinite-dimensional?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this? | 868858 | Dimension of R over Q without cardinality argument.
I am looking for the easiest (elementary) proof that [imath]\mathbb R[/imath] is infinite dimensional as a [imath]\mathbb Q[/imath]-vector space, without using cardinality. It should be understandable at highschool level. So I guess the question could be reformulated... |
147642 | If a group satisfies [imath]x^3=1[/imath] for all [imath]x[/imath], is it necessarily abelian?
I know that any group satisfying [imath]x^2=1[/imath] for all [imath]x[/imath] is abelian. Is the same true if [imath]x^3=1[/imath]? I don't think it is, but I can't find a basic counterexample. | 678570 | Non-abelian group in which [imath]\forall_{a,b\in G} (ab)^3=a^3b^3[/imath]
Give an example of a non-abelian group, in which [imath](ab)^3=a^3b^3[/imath] for every element [imath]a,b[/imath] in [imath]G[/imath]. I understand that such a group should be of order divisible by 3 (see Problem from Herstein on group theory)... |
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