1. sum series

Some han help to calculate the serie
$\sum _{z=1}^{\infty } \frac{(-1)^{-1+z}}{\left(-\frac{4}{3}+\sqrt{2+z}\right)^3}$
thanks

2. Re: sum series

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

3. Re: sum series

Originally Posted by Prove It
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

4. Re: sum series

$\frac{13095}{121}+\frac{81}{\sqrt{2}}-\frac{243 \sqrt{3}}{121}-2 \text{PolyGamma}\left[1,\frac{1}{9}\right]+2 \text{PolyGamma}\left[1,\frac{11}{18}\right]-\frac{32}{27} \text{PolyGamma}\left[2,\frac{11}{18}\right]+\frac{32}{27} \text{PolyGamma}\left[2,\frac{10}{9}\right]$
working on it I found an aproximate sum

5. Re: sum series

Originally Posted by capea
$\frac{13095}{121}+\frac{81}{\sqrt{2}}-\frac{243 \sqrt{3}}{121}-2 \text{PolyGamma}\left[1,\frac{1}{9}\right]+2 \text{PolyGamma}\left[1,\frac{11}{18}\right]-\frac{32}{27} \text{PolyGamma}\left[2,\frac{11}{18}\right]+\frac{32}{27} \text{PolyGamma}\left[2,\frac{10}{9}\right]$
working on it I found an aproximate sum
If that is an approximation how is it more useful than say: 13.3083968 +/- 0.0000359 ?

CB

6. Re: sum series

Thanks for your comments on the previous approximation expressions lack a term 0(x^-4), but i have not clue how to calculate any idea ?

7. Re: sum series

Originally Posted by capea
Thanks for your comments on the previous approximation expressions lack a term 0(x^-4), but i have not clue how to calculate any idea ?
What is wrong with summing 100000 terms of the series?

CB

8. Re: sum series

Originally Posted by capea
Thanks for your comments on the previous approximation expressions lack a term 0(x^-4), but i have not clue how to calculate any idea ?
What would x be? You have not told us how you obtained your approximation so we have no idea what x is here.

CB