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.