How to solve these series? Can anyone help me understand how to solve these two series?  More than the solution I'm interested in understanding which process I should follow.


*

*Series 1:


$$
 \sum_{i = 3}^{\infty} i * a^{i-1}, 0 < a < 1.
$$


*

*Series 2:


$$
 \sum_{i = 3}^{\infty} i\sum_{k = 2}^{i-1} a^{i-k} * b^{k-2} , 0 < a < 1, 0 < b < 1.
$$
These two series come as part of a long mathematical proof which I omitted for brevity, if you think it is relevant I will post it.

Hint: For the first one if $\sum_{i = 3}^{\infty} x^{i}=f(x)$ then $\sum_{i = 3}^{\infty} i \times x^{i-1}=f'(x)$.

For the second one consider that:
$$
\sum_{k = 2}^{i-1} a^{i-k}  b^{k-2}=a^{i-2}\frac{1-\frac{b^{i-2}}{a^{i-2}}}{1-\frac{b}{a}}= \frac{a^{i-2}-{b^{i-2}}}{1-\frac{b}{a}} 
$$
Then you can decompose the series into two one and use the previous step.
