# Convergence

#### Alexmahone

##### Active member
Does the following series converge?

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

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#### fawaz

##### New member
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

#### Alexmahone

##### Active member
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
$\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty}\frac{(n+1)^5}{2n^5}=\frac{1}{2}$

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

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#### Krizalid

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