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Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum: $$ f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...

On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula $$\sum_{k=0}^{n-1}(-1)^k\cot\...

After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero: $$\lim_{N\to\infty}\frac{1}{N}\sum_{m=1}^\infty\frac{...

The copy of this question is posted here I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{ ...

Consider the random variable $$X=\frac{1}{d}\sum_{k=1}^d\cos X_k$$ where $X_k$ are each drawn uniformly i.i.d. from $[0,2\pi]$. What is the CDF of X? It seems that a relatively direct way could be to ...