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#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

Does the following series converge?

$\displaystyle\sum\frac{n^5}{2^n}$

$\displaystyle\sum\frac{n^5}{2^n}$

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- Thread starter Alexmahone
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- Thread starter
- #1

- Jan 26, 2012

- 268

Does the following series converge?

$\displaystyle\sum\frac{n^5}{2^n}$

$\displaystyle\sum\frac{n^5}{2^n}$

Last edited:

- Thread starter
- #3

- Jan 26, 2012

- 268

$\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty}\frac{(n+1)^5}{2n^5}=\frac{1}{2}$Well what tests did you try?the ratio test and the root test both give you what you need. Try them. If you need more hints let us know.Mohammad

So, $\displaystyle\sum\frac{n^5}{2^n}$ converges.

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- Feb 9, 2012

- 118

$a_n=\dfrac{n^k}{2^n},$ so by the root test the series always converges for all $k.$