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Question: Find $\displaystyle \sum \limits_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}$

My issue(s): How do I do this without using a calculator? I know that I have to do something with $S_n$ and $S_{n+1}$ but I'm not sure what.

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up vote 9 down vote accepted

Use properties of exponents to rewrite $$\frac{2^{2n+1}}{5^n}$$ in the form $$A(B)^n.$$ You probably recognize that form as a geometric sequence/series. The sum of an infinite geometric series with first term $a_1$ and constant ratio $r$ with $-1<r<1$ is $$\frac{a_1}{1-r}.$$

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HINT Try to write this as a geometric series by pulling out a factor of $2$ and rewriting $2^{2n}$ as $4^n$

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Let $\displaystyle{S=\sum_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}=\sum_{n=1}^{\infty} 2(\frac{4}{5})^n}$.

Then, $S=2(\frac{4}{5}+\frac{16}{25}+\frac{64}{125}+...)$

And, $\frac{4S}{5}=2(\frac{16}{25}+\frac{64}{125}+\frac{256}{625}+...)$

Now, what happens if you subtract the second equation from the first?

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do not pull the $2$ out of each summand, but out of the whole sum. Generally, try get "wild" expressions as clear as possible from non essential stuff. And now learn about the geometric series in Wikipedia. – shuhalo Mar 21 '11 at 0:06

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