Bug introduced in 7.0 and fixed in 9.0.0
According to the documentation
GeneratingFunction[a[n],n,x]==Sum[a[n]x^n,{n,0,Infinity}]
However, for $a_n=1/(n+2)$ I obtain
{Sum[1/(n + 2) x^n, {n, 0, Infinity}], GeneratingFunction[1/(n + 2), n, x]} // FullSimplify
% /. x -> .2
(*{-((x + Log[1 - x])/x^2), PolyLog[2, x]/x}*)
(*{0.578589, 1.05502}*)
Thanks
Edit:
I'm using Mathematica 8.0.0.0 on a Linux x86 (32-bit)
GeneratingFunction[1/(n + 2), n, x] ==> (-1-(Log[1-x]/x))/x
$\endgroup$ – István Zachar Sep 3 '13 at 15:41GeneratingFunction[1/(n + 2), n, x]
? $\endgroup$ – Mr.Wizard Sep 3 '13 at 15:42(-1 - Log[1 - x]/x)/x
in v9.PolyLog[2, x]/x
in v8. $\endgroup$ – Szabolcs Sep 3 '13 at 15:44