# Tagged Questions

Convergence of sequences and different modes of convergence.

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### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ 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 ...
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### How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
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### Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational

Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
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### “Why do I always get 1 when I keep hitting the square root button on my calculator?”

I asked myself this question when I was a young boy playing around with the calculator. Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman ...
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### Is there a way of working with the Zariski topology in terms of convergence/limits?

As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...
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### A double sum $\sum \limits_{n=1}^{n=\infty}\left(\sum \limits_{k=n}^{k=n^2}\frac{1}{k^2}\right)$

How to evaluate $\displaystyle\sum_{n=1}^{n=\infty}\left(\sum_{k=n}^{k=n^2}\frac{1}{k^2}\right)$?
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### Show that the sequence ${a_n}$ converges where $a_n = \sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{n}}}}$ for $n\geq 1$.

The original question was to determine whether the sequence converges, but I have checked for extremely high values of $n$ and it seems as though it does converge. This lead me to wonder if there was ...
### Strong and weak convergence in $\ell^1$
Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...