Welcome to our community

Be a part of something great, join today!

Convergence

Alexmahone

Active member
Jan 26, 2012
268
Does the following series converge?

$\displaystyle\sum\frac{n^5}{2^n}$
 
Last edited:

fawaz

New member
Feb 17, 2012
3
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
Jan 26, 2012
268
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.
 
Last edited:

Krizalid

Active member
Feb 9, 2012
118
$a_n=\dfrac{n^k}{2^n},$ so by the root test the series always converges for all $k.$