2005 Alabama ARML TST Problems/Problem 7

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Problem

Find the sum of the infinite series:

$3+\frac{11}4+\frac 94 + \cdots + \frac{n^2+2n+3}{2^n}+\cdots$.

Solution

$\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}=\sum_{n=1}^{\infty} \left(\frac{n^2}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{2n}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)$

We can compute those sums:

$\begin{eqnarray*} \sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)=x\\ =3\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\right)\\ 2x=3\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\right)\\ x=3(1)=3\\ \sum_{n=1}^{\infty} \left(\frac{2n}{2^n}\right)=y\\ =\frac{2}{2}+\frac{4}{4}+\frac{6}{8}+\frac{8}{16}+\cdots\\ 2y=2+\frac{4}{2}+\frac{6}{4}+\frac{8}{8}+\cdots\\ y=2+\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\cdots=4\\ \sum_{n=1}^{\infty} \left(\frac{n^2}{2^n}\right)=z\\ =\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\cdots\\ 2z=1+\frac{4}{2}+\frac{9}{4}+\frac{16}{8}+\cdots\\ z=1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\frac{9}{16}+\cdots\\ 2z=2+3+\frac{5}{2}+\frac{7}{4}+\frac{9}{8}+\cdots\\ z=4+\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\cdots=6\\ 3+4+6=\boxed{13} \end{eqnarray*}$

If you didn't understand the third case / summation, all they did was subtract the summation for $z$ from $2z$ to get the new $z$ and then repeat this.