I need help with understanding how to solve this task, because I'm a bit lost at the moment.
Use the powerseries $$f(x)=\frac{1}{1-x}$$ to decide the sum of the series
$\sum_{n=1}^{\infty} n(n+1)x^n$ and $\sum_{n=1}^{\infty} \frac{n(n+1)}{3^n}$
I don't understand how to manipulate the sums to use the power series of the function.
First, note that for $|x| < 1$ we have $f(x) = \sum_{n=0}^\infty x^n$.
and for each $|r|<1$ the series converge uniformly in $[-r,r]$ so we can use "derivative term by term" to get:
$f'(x) = \sum_{n=0}^\infty nx^{n-1}$. Use this reasoning again to get
$f''(x) = \sum_{n=0}^\infty (n-1)nx^{n-2}$
Use this to calculate $\sum_n n(n+1)3^{-n}$