2017-20 Convergence of a series

Determine whether or not the following infinite series converges. $\sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.$

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2017-19 Identity

For an integer $$p$$, define
$f_p(n) = \sum_{k=1}^n k^p.$
Prove that
$\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.$

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2016-12 A series

Let $$k$$ be a positive integer. Let $$a_n=1$$ if $$n$$ is not a multiple of $$k+1$$, and $$a_n=-k$$ if $$n$$ is a multiple of $$k+1$$. Compute $\sum_{n=1}^\infty \frac{a_n}{n}.$

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2014-07 Subsequence

Let $$a_1,a_2,\ldots$$ be an infinite sequence of positive real numbers such that $$\sum_{n=1}^\infty a_n$$ converges. Prove that for every positive constant $$c$$, there exists an infinite sequence $$i_1<i_2<i_3<\cdots$$ of positive integers such that $$| i_n-cn^3| =O(n^2)$$ and  $$\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)$$ converges.

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2014-02 Series

Determine all positive integers $$\ell$$ such that $\sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}$ converges and if it converges, then compute its value.

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2012-17 Two Tangent Functions in a Series

Let $$m$$ and $$n$$ be odd integers. Determine $\sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.$

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2011-1 A Series

Evaluate the sum $\sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}.$

(UPDATED: 2011.2.18) I have fixed a typo in the formula. Initially the following formula $\sum_{n=1}^{\infty} \frac{\sin n}{1+n^2}$ was posted but it does not seem to have a closed form answer. I’m sincerely sorry!

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2010-3 Sum

Evaluate the following sum

$$\displaystyle\sum_{m=1}^\infty \sum_{\substack{n\ge 1\\ (m,n)=1}} \frac{x^{m-1}y^{n-1}}{1-x^m y^n}$$

when |x|, |y|<1.

(We write (m,n) to denote the g.c.d of m and n.)

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$$\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}$$