# What programs can calculate (this) series (get expression in closed form)?

What programs can calculate this (type of) series?

$$\sum_{m,n=0}^\infty \frac{(-1)^{-2n}2^{-2m-n}\,(1+m)}{2+m+n}$$

One program I know of is XSummer math-ph/0508008, XSummer.

The result is:

$$-16\cdot \left(\frac{1}{3}-\log\frac{3}{2}\right)$$

Because I will need to calculate more complicated series (4+ fold) of the same form (also resulting in log2 and log3 terms) I'd like to get a decent overview. Common CAS like maple or mathematica can perform just one series, resulting in hypergeometric expressions, for which they cannot perform the second series summation.

## 1 Answer

I just typed that into mathematica and got $\frac{8}{3} \left(\log \left(\frac{729}{64}\right)-2\right)$.

Image as follows: • D'oh, alright. I did not know mathematica was able to compule such multi-series in one go. I used Sum[Sum[...,{m,0,Infinity}],{n,0,Infinity}]. Well, I guess I have to come up with a harder example that doesn't work with mathematica :) We can still collect other software though. – Tobias May 24 '13 at 21:03
• @Tobias just going back to this, something called the (Wolfram Language)[wolfram.com/language/] is coming out which I'm guessing will have all the algorithms for this built in. Hope that helps! – DanZimm Apr 17 '14 at 5:36