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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}...

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)\\...

See Also

2005 Alabama ARML TST (Problems)
Preceded by:
Problem 6 		Followed by:
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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