I need to find the sum of this series: $1, 2 \left ( 1 - \frac{1}{\sqrt{15}} \right ), 3 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 2, 4 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 3, 5 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 4, ... $
I easily found the formula: $\displaystyle\sum_{n=0}^{\infty} (n+1) \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ n$
but now I don't know how to find the solution without using limits. I know I could solve the limit: $\displaystyle\lim_{k\to\infty}\sum_{n=0}^{k} (n+1) \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ n$
I was wondering if there is a way to solve the sum without using limits.