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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. ...

Here Gottfried Helms introduces the following fascinating divergent series $$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$ The terms don't go to zero, so technically the series does not converge ...

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 ...

My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...