# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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### Let [$c_n$]{for $n$ from $0$ to $+\infty$} be a sequence of real numbers.

It is known, that $c_n>1$ for any $n$ and that $\prod_{i=0}^\infty c_i$ diverges to $\infty$ Is it true that $\sum_{i=0}^\infty ((1/c_i)-(1/(c_ic_{i+1}))$ also diverges to infinity?
1answer
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1answer
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### How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$

How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$ Here's my progress: I suppose $\sum_{n,m=2}^{\infty}{n^{-m}}=\sum_{n=2}^{\infty}{\sum_{m=2}^{\infty}{\left(\frac{1}{n}\right)^m}}$, so we're looking at ...
1answer
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### Conditions for Differentiable Limit Theorem

I know that if $(f_n)$ is a sequence of differentiable functions on $(a,b)$ with pointwise limit $f$ and if $f'_n \rightarrow g$ uniformly the $f$ is differentiable on $(a,b)$ and $f' = g$. I want to ...
1answer
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### A sequence $x_n$ has a cauchy subsequence ii it has a subsequence satisfying the following property [on hold]

I am trying to solve this problem but could not make an idea. Please give some hint for the problem.
3answers
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### Find the sum of the series $1^3 + 3\cdot 2^2 + 3^3 + 3\cdot 4^2 + 5^3 + 3\cdot 6^2…$ up to $n$ terms

Find the sum of first $n$ terms of the series $1^3 + 3\cdot 2^2 + 3^3 + 3\cdot 4^2 + 5^3 + 3\cdot 6^2...$ When $n$ is even. When $n$ is odd. This sum can be written as \sum_{1}^n (...
5answers
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### $\sum_{n=0}^{\infty} (n+1)^2 x^n$ Closed Form [duplicate]

$\sum_{n=0}^{\infty} (n+1)^2 x^n$ Closed Form I'm a bit stuck on finding the closed form here. I don't think I can use the technique of computing derivatives here directly. Could someone point me in ...
5answers
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### $\sum_{n=0}^{\infty} (2n+1)x^n$ Closed Form

$\sum_{n=0}^{\infty} (2n+1)x^n$ Closed Form I'm trying to find the closed form for the specified series. However, I'm having a bit of trouble doing so. I assume there's a technique here that I haven'...