Summation #4

Relevant wiki: Geometric Progression Sum

$\begin{aligned} S & = \sum_{n=0}^\infty \frac {n^22^{n-1}}{(-2)^{2n}} \\ & = \sum_{\color{#3D99F6}n=0}^\infty \frac {n^2}{2^{n+1}} = \sum_{\color{#D61F06}n=1}^\infty \frac {n^2}{2^{n+1}} \\ & = \sum_{\color{#3D99F6}n=0}^\infty \frac {(n+1)^2}{2^{n+2}} \\ & = \frac 12 \sum_{n=0}^\infty \frac {n^2+2n+1}{2^{n+1}} \\ & = \frac 12 \sum_{n=0}^\infty \frac {n^2}{2^{n+1}} + \frac 12 {\color{#3D99F6} \sum_{n=0}^\infty \frac n{2^n}} + \frac 14 \sum_{n=0}^\infty \frac 1{2^n} & \small \color{#3D99F6} \text{See note.} \\ & = \frac 12 S + \frac 12 {\color{#3D99F6} \sum_{n=0}^\infty \frac 1{2^n}} + \frac 14 \sum_{n=0}^\infty \frac 1{2^n} \\ & = \frac 32\color{#3D99F6} \sum_{n=0}^\infty \frac 1{2^n} & \small \color{#3D99F6} \text{Sum of infinite geometric progression} \\ & = \frac 32 \color{#3D99F6} \left(\frac 1{1-\frac 12}\right) \\ & = \boxed 3 \end{aligned}$

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Note:

$\begin{aligned} S_1 & = \sum_{\color{#3D99F6}n=0}^\infty \frac n{2^n} = \sum_{\color{#D61F06}n=1}^\infty \frac n{2^n} \\ & = \sum_{\color{#3D99F6}n=0}^\infty \frac {n+1}{2^{n+1}} \\ & = \frac 12 \sum_{n=0}^\infty \frac n{2^n} + \frac 12 \sum_{n=0}^\infty \frac 1{2^n} \\ & = \frac 12 S_1 + \frac 12 \sum_{n=0}^\infty \frac 1{2^n} \\ & = \sum_{n=0}^\infty \frac 1{2^n} \end{aligned}$