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 $k.$