I'm defining the following function $S(x,n)=\sum_{i=1}^\infty i^nx^i$
Just to list the first few,
$S(x,1)=\frac{x}{(1-x)^2}$
$S(x,2)=\frac{x(x+1)}{(1-x)^3}$
$S(x,3)=\frac{x(x^2+4x+1)}{(1-x)^4}$
I tried to evaluate $S(x,\frac{1}{2})$ but am getting no where. I've done the above three examples with algebra. There's probably some function or theorem that can solve this but I have no idea.
I'd like the function S(x,1/2) to be defined for all real value of x except x=1, for that will give the function $\sqrt 1 +\sqrt 2 +\sqrt 3...$ which is divergent
Any advise is appreciated.
P.S. This isn't for paper or assignment so don't worry.
Basically, what you are looking for is the polylogarithm function $$S(x,n)=\sum_{i=1}^\infty i^nx^i=\text{Li}_{-n}(x)$$ which cannot reduce to anything is $n$ is not an integer $\geq -1$.
So, to add to the ones you already have $$S(x,0)=-\frac{x}{x-1}$$ $$S(x,-1)=-\log (1-x)$$ All other invoke the special function.