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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 ...