# Tagged Questions

Convergence of sequences and different modes of convergence.

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${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad$ in probability show that $$\lim_{n\to \infty} \min_{... 3answers 57 views ### convergence of \int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx How do I prove convergence of$$\int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx$$and if it's convergent, calculate the value of the integral? I noticed that the values that the function ... 1answer 40 views ### Where is the mistake of a possible application of Frullani's theorem in this case? My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ... 1answer 36 views ### Convergence of the integral: I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx Study the convergence of the integral:$$I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$$and calculate I_2. Ok so to study the convergence I'm using convergence ... 3answers 681 views ### Convergence of the series \sum_{n=1}^{\infty}((1/n)-\sin(1/n))..? Does the series \sum_{n=1}^{\infty}((1/n)-\sin(1/n)) converge..? Can anyone please give me a simple proof.. 2answers 43 views ### How to show the sequence is monotone "u_n = \frac{2}{1+e^{-n}}. Show that u_n is monotone." My approach would be to consider |u_{n+1} - u_n| = |\frac{2}{1+e^{-n-1}} - \frac{2}{1+e^{-n}}|. However I'm not sure the best way to ... 1answer 22 views ### Convergence/Divergence of Integral, can P-test be used here? I have an integral like this: How do I check its convergence? As far as I know, P-test can be used for integrals from 0 to 1, or A to infinity, what would I do in this case? 1answer 35 views ### An ancillary result from convergence in probability I was reading a paper concerning probability theory. We have that X_i, i = 1,2,... i.i.d random variables, and S_n = \sum_{i=1}^nX_i is defined as partial sum as usual. If$$\frac{S_n}{n} \...
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I'm pretty sure I'm missing something very basic... But I have the following question: Determine the radius of convergence of $\sum \alpha_n z^n$ with $\alpha_n=\frac{1}{n+1}$. Now, with the ratio ...
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### Convergence of integral with absolute function

Given that the following integral converges: $$\int_{0}^\infty |f(x)| dx$$ Prove or disprove that the following integral also converges: $$\int_{0}^\infty f(x) dx$$ I thought to use the squeeze ...
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### Fourier series for $\min \{0, \cos x\}$

How can I find Fourier series and convergence of function $f(x)= \min \{0, \cos x \}$ ? Because it is an even function I am expanding it only for cosine. Doing that I get $a_{0} = 0$, after that I ...
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### What should be the value of $\alpha$ for which the series is convergent?

The series $$\sum \frac{\log(1+\frac{1}{n})}{n^\alpha}$$ a. Converges if $\alpha>0$ b. Diverges for all $\alpha\in \mathbb{R}$ c. Converges if $\alpha=0$ d. Converges if $\alpha<0$ ...
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### (Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
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I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of $a$ does the following integrals exists $$(1) \int_0^\infty\... 1answer 71 views ### Convergence of the integral \int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx as N\to\infty How can calculate this integral$$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$where f(x) is a probability density function?... 1answer 26 views ### weakly convergence the sequence f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]} I need to research on the uniform, weak and strong convergence the sequence$$f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$$for n\in \mathbb{N}, in L^{2}(\mathbb{R}) equipped with norm \... 2answers 35 views ### Comparision test for this series? How do I check divergence of this series?$$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ... 1answer 50 views ### conditional convergence of \sum_{n=2}^{\infty} \frac{\cos(n)}{n} prove that the series$$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that \sum_{n=2}^{\infty} \... 1answer 66 views ### Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables? I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence (X_n)_{n \in \mathbb{N}} of non-negative i.i.d. RV X_n \sim X... 1answer 48 views ### Prove A_{\infty} < \infty? From Williams' Probability with Martingales How do we know that A_{\infty} < \infty? If T = \infty, then$$E[A_{T \wedge n}] \le (K+c)^2\to E[A_{n}] \le (K+c)^2\to \lim ...
Don't mark this as duplicate. The question is to verify whether my method is correct. Prove: $$\lim_{n\to\infty} \frac{x^n}{n!} = 0$$ Method: Let $\sum_{n=1}^{\infty} \frac{x^n}{n!}$. By ratio ...
What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that \$...