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I have the summation

$$ \sum_{n = 0}^{10} \frac {1 + (-1)^n} {2^n} $$

I have looked up how to work out sequences without manually finding each term and adding it up, but I have only found out how to work out problems like $n^2$, $n^3$, etc.

How would I solve this? Is there some way to simplify it to be able to solve it? Please be detailed.

Thanks

Hagen von Eitzen
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Maria
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    The numerator is $0$ when $n$ is odd, and $2$ when $n$ is even. So we get $2\left(1+\frac{1}{2^2}+\frac{1}{2^4}+\cdots+\frac{1}{2^{10}}\right)$. Now we have a finite geometric series, common ratio $\frac{1}{4}$. – André Nicolas Oct 23 '13 at 18:23

2 Answers2

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Finite sums are, well, finite. So we may simply expand $$ \begin{align}\sum_{n=0}^{10}\frac{1+(-1)^n}{2^n}&=\frac{1+(-1)^0}{2^0}+\frac{1+(-1)^1}{2^1}+\frac{1+(-1)^2}{2^2}+\ldots +\frac{1+(-1)^{10}}{2^{10}}\\ &=\frac21+0+\frac24+0+\frac2{16}+0+\frac2{64}+0+\frac2{256}+0+\frac2{1024}\\ &=\frac{1024+256+64+16+4+1}{512}=\frac{1365}{512}\end{align}$$

Hagen von Eitzen
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$$\sum_{0\le n\le10}\frac{1+(-1)^n}{2^n}=\sum_{0\le m\le5}\frac2{2^{2m}}$$ which is a Geometric Series

lab bhattacharjee
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