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When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...

Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...

Cross-posted from Math SE. The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question: ...

I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...

Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$, $$ \log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}. $$ It is well known, ...