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What is the current state of the art in summing (where by 'summing', I mean 'representing in terms of already known constants and whatnot') series such as these:

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{1+7^{n}}}$$

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}$$

$$\sum_{n=1}^{\infty} e^{-\sqrt{n}}$$

$$\sum_{n=1}^{\infty} \frac{1}{n^{3}+\sqrt[7]{n}}$$

I have a copy of Konrad Knopp's book /Theory and Application of Infinite Series/ , but that's fifty years old, and I've been hoping that there have been improvements in the techniques since then.

3 Answers 3