# Find sum of power series $\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n$

I have calculate the sum of the following power series: $$\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n$$

I've determined the convergence radius $R = 1$convergence interval $I_c = (-1,1)$ and the convergence set $A = (-1,1)$ for this power series. Furthermore, I've tired to bring it to a from which can be easily computed, namely $$\sum_{i=0}^{\infty}x^n$$ or $$\sum_{i=0}^{\infty}(-1)^n x^n$$ (I only guess that these are the closest nicest forms that can be reached).

I tried to create a new function $g(x)=\int f(x) \, dx = \sum_{n=0}^{\infty}(-1)^n(n+1)x^{(n+1)}$, where $f(x)$ is the function in the power series at the very beginning of this question.

And this is where I get stuck. Any help would be much appreciated.

• Look at $g(x)/x$. Commented Jun 20, 2017 at 21:58

Recall that $F(x)=\frac{1}{1+x}=\sum_{n=0}^\infty (-1)^n x^n$. Differentiating, we get $$F'(x)=\sum_{n=0}^\infty (-1)^n n x^{n-1}=-\sum_{n=0}^\infty(-1)^{n}(n+1)x^n.$$ Differentiating again: $$F''(x)=\sum_{n=0}^\infty (-1)^{n+1}(n+1)n x^{n-1}=\sum_{n=0}^\infty(-1)^{n}(n+2)(n+1)x^n.$$ Thus, $$F''(x)+F'(x)=\sum_{n=0}^\infty(-1)^n(n+1)[(n+2)-1]x^n=\sum_{n=0}^\infty (-1)^n(n+1)^2x^n=f(x),$$ which recovers your sum after you evaluate the left hand side.
We have the following geometric series \begin{eqnarray*} \sum_{n=0}^{\infty} (-1)^n x^n = \frac{1}{1+x} \end{eqnarray*} Now multiply this by $x$ and differentiate \begin{eqnarray*} \sum_{n=0}^{\infty} (-1)^n (n+1) x^n = \frac{(1+x)-x}{(1+x)^2}=\frac{1}{(1+x)^2} \end{eqnarray*} & now do exactly the same again to this & we have \begin{eqnarray*} \sum_{n=0}^{\infty} (-1)^n (n+1)^2 x^n = \frac{1}{(1+x)^2}-\frac{2x}{(1+x)^3}=\color{red}{\frac{1-x}{(1+x)^3}}. \end{eqnarray*}
First, let $x=-y$ which makes $$S=\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n=\sum_{n=0}^{\infty} (n+1)^2y^n=\sum_{n=0}^{\infty} (n^2+2n+1)\,y^n$$ Now, use $n^2=n(n-1)+n$ which makes $$S=\sum_{n=0}^{\infty} (n(n-1)+3n+1)\,y^n$$ that is to say $$S=\sum_{n=0}^{\infty} n(n-1)\,y^n+3\sum_{n=0}^{\infty} n\,y^n+\sum_{n=0}^{\infty} y^n$$ $$S=y^2\sum_{n=0}^{\infty} n(n-1)\,y^{n-2}+3y\sum_{n=0}^{\infty} ny^{n-1}+\sum_{n=0}^{\infty} y^n$$ $$S=y^2\left(\sum_{n=0}^{\infty} y^n \right)''+3y\left(\sum_{n=0}^{\infty} y^n \right)'+\left(\sum_{n=0}^{\infty} y^n \right)$$ I am sure that you can take it from here.