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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: ...

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(...