I am trying to find an asymptotic formula for the sum of the following:
$\sum _{x=1}^{\infty } x \left(\left(1-\frac{\Gamma (x,\lambda )}{\Gamma (x)}\right)^n-\left(1-\frac{\Gamma (x+1,\lambda )}{\Gamma (x+1)}\right)^n\right)$
I am new to this and any direction would be appreciated.