Some han help to calculate the serie
thanks
Wolfram can't get an exact answer, but the series is convergent by the Alternating Series Test.
http://www.wolframalpha.com/input/?i=sum+of+%28-1%29^%28z-1%29%2F%28Sqrt%282%2Bz%29-4%2F3%29^3+with+z+from+1+to+infinity
What I find amusing about the Alpha attempt to sum this series is that it reports negative results for a couple of convergence tests, and ignores the fact that it is obviously convergent (in fact it is obviously absolutly convergent).
(Note I asked for a simplified rearrangenment of this which it failed to notice converged by the alternating series convergence test)
CB