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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.

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3 Answers

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.

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I've seen this before, and I like Plouffe's transformation, but the zeta function is kind of a rock star, so is going to get a lot of attention. –  deoxygerbe Oct 15 '10 at 11:34
    
On a related note, has anyone figured out what symmetry is responsible for the "neat" even values of $\zeta(s)$ and the "messy" odd values? –  deoxygerbe Oct 15 '10 at 11:35
    
The reason for the dichotomy between even/oddness for the zeta values is that the standard method computes $\sum_{n\in\mathbb{Z},n\ne0}n^{-k}$ for each integer $k\ge2$. And it does so successfully whether or not $k$ is even or odd. –  Robin Chapman Oct 15 '10 at 15:48
    
@Robin: the zeta function of even values is well known, the odd (via the Plouffe method above) isn't as well known. More to the point, the zeta function of even numbers has a nice compact form, but the odd ones are rather odd. I was kind of hoping that there was an interpretation of this phenomena in terms of the $1/\zeta(s)$ being the probability that $s$ randomly chosen integers were coprime -- something about choosing an even number of random integers is more symmetrical than choosing an odd number of random integers. But I don't have anything more than that speculation. –  deoxygerbe Oct 15 '10 at 23:29
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You may want to read up on Gosper's algorithm. It helps you find closed-form expressions for certain classes of sums.

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Most of the sums that I end up dealing with aren't hypergeometric, e.g. monsters like $\sum_{a,b,c=1}^{\infty} \frac{1}{(abc)^{abc}}$ –  deoxygerbe Mar 1 '11 at 16:56
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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.

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