Originally Posted by

**CaptainBlack** Put in sufficient brackets to make your expressions unambiguous!, As it stands there is no closed form as the inner sum is divergent.

So let us assume you mean:

$\displaystyle S(n)=\sum_{j=1}^n \sum_{i=0}^{\infty} j^{5/3}\left(1-\frac{1}{2j^{1/3}}\right)^i$

Write it as:

$\displaystyle S(n)=\sum_{j=1}^n j^{5/3} \sum_{i=0}^{\infty} \left(1-\frac{1}{2j^{1/3}}\right)^i$

Now the inner sum is an infinite geometric series with sum:

$\displaystyle S_{inner}(j)= \frac{1}{1-(1-1/(2j^{1/3}))}=2j^{1/3}$

...

CB