function to which power series converges within radius of convergence

Question

The question is to determine the radius of convergence of as well as the function to which the following series converges to (within that radius): $\sum_{j=1}^{\infty} (j + 1)(j + 2)x^j$.

I was able to determine that it converges whenever $|x| < 1$, but I wasn't able to determine to what function it will converge (or think of a starting point for that). Any tips?

Answer 1

Doing integration may help: \begin{align*} \int_{0}^{x}\sum_{j=1}(j+1)(j+2)t^{j}dt&=\sum_{j=1}(j+2)t^{j+1}\bigg|_{t=0}^{t=x}\\ &=\sum_{j=1}(j+2)x^{j+1}, \end{align*} and once more \begin{align*} \int_{0}^{x}\sum_{j=1}(j+2)t^{j+1}dt=\sum_{j=1}x^{j+2}=\dfrac{x^{3}}{1-x}, \end{align*} so the series is the twice derivative of $x^{3}/(1-x)$.

Answer 2

Think about derivatives of geometric series: $$\sum_{j=1}^{\infty}x^{j+2}=\frac{x^3}{1-x}$$

around $|x|<1$.