Interval of Convergence of the following...
Under the hypothesis that is...
$\displaystyle f(x)= \sum_{k=1}^{\infty} k a^{k} x^{k}$ (1)
... setting $\displaystyle a x = \xi$ we have...
$\displaystyle \sum_{k=1}^{\infty} k a^{k} x^{k}= \sum_{k=1}^{\infty} k \xi^{k}= \xi \sum_{k=1}^{\infty} k \xi^{k-1} = \xi \frac{d}{d \xi} \sum_{k=1}^{\infty} \xi^{k} = $
$\displaystyle = \xi \frac{d}{d \xi} \frac{\xi}{1-\xi} = \frac{\xi}{(1-\xi)^{2}} $ (2)
The series converges for $\displaystyle |\xi|<1$ ...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$