what is the current state of the art in methods of summing "exotic" series?

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.

Solution 1:

Well, much depends on the individual series but one interesting development in the last 50 years is on the acceleration of convergence, which enables us to express some slowly convergent series in terms of one(s) with much faster convergence, or a sum of known constants plus faster converging series. For example, I really like this transformation due to Simon Plouffe

$$\zeta(7) = \frac{19}{56700}\pi^7 – 2\sum_{n=1}^{\infty} \frac{1}{n^7(e^{2\pi n}-1)},$$

where $\zeta(7) = \sum_ {n=1}^{\infty} 1/n^7,$ which is given on this wikipedia page.

Andrei Markov's 1890 method, which was the basis for Apéry's proof of the irrationality of $\zeta(3),$ has been pushed further by Mohammed and Zeilbeger.

You can see some other nice transformations of $\zeta(3)$ here.

Solution 2:

You may want to read up on Gosper's algorithm. It helps you find closed-form expressions for certain classes of sums.

Solution 3:

Take a look at Petkovsek, Wilf and Zeilberger's book "A = B". They tame an entire zoo of sums. Also of interest is Wilf's "generatingfunctionology", his "snake oil method" is simple to apply by hand and works in a surprising variety of situations.