# Tagged Questions

Convergence of sequences and different modes of convergence.

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### If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
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### Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Show that $$\int\limits^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$$ converge. I have utterly no clue on this integral. Please give me some hints. Thanks you.
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### Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
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### Sufficiency to prove the convergence of a sequence using even and odd terms

Given a sequence $a_{n}$, if I know that the sequence of even terms converges to the same limit as the subsequence of odd terms: $$\lim_{n\rightarrow\infty} a_{2n}=\lim_{n\to\infty} a_{2n-1}=L$$ ...
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### Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
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### The series $\sum a_n$ is conditionally convergent. Prove that the series $\sum n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $|a_n|$ is not necessarily smaller than $n^2a_n$. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0$ ...
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### Convergence of $\sum_n \frac{n!}{n^n}$

I'm working on a problem sheet and it ask to discuss the convergence of $$\sum \frac{n!}{{n}^{n}}$$ By D'Lembert's ratio test, $$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$ and so, is ...
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### A sequence converges if and only if every subsequence converges?

I want to prove this and intuitively it makes sense. But I'm having a hard time coming up with a proof. So if a sequence converges, then we have a natural number for which the distance between all ...
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### How to calculate $\sum_{n=0}^\infty {(n+2)}x^{n}$

I want to calculate the sum of $$\sum_{n=0}^\infty {(n+2)}x^{n}$$ I have tried to look for a known taylor/maclaurin series to maybe integrate or differentiate...but I did not find it :| Thank you. ...
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### What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow X$,...
### Convergence of $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$
I'm trying to prove the convergence of $$\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$$ with $\alpha > 1$. For $\alpha \geq 2$ I can use the comparison test ($\sum_{n=1}^{\infty} \frac{1}{n^2}$ ...