My book has writen this. Series is attached
How he has converged the series
That's almost a geometric series. There is a general formula that
$\displaystyle \sum_{n=0}^\infty ar^n= \frac{a}{1- r}$.
Now, the derivative, term by term, is $\displaystyle \sum_{n=0}^\infty a_n r^{n-1}$ and that must be equal to the derivative, with respect to r, of $\displaystyle \frac{a}{1- r}$.