All Questions

Background A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...

Background Taking a relatively arbitrary combination of exponential and polynomial terms, for instance $$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...

So I was considering the divergent everywhere but 0 power series $$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$ Now one can do the following "questionable" manipulation $$ f(x) = \sum_{n=0}^{\...