# determine converges absolutely, conditonally or if diverges

• Jan 30th 2007, 06:47 PM
rcmango
determine converges absolutely, conditonally or if diverges
heres the equation: http://img440.imageshack.us/img440/6467/untitledbw7.jpg

I showed that it was an alternate series, and i predicted that it converges absolutely, however i'm missing steps.

help is needed.
• Jan 30th 2007, 07:08 PM
Soroban
Hello, rcmango!

Quote:

Heres the series: .$\displaystyle \sum^{\infty}_{n=0}\frac{(-1)^nn}{\sqrt{1 + n^6}}$

I showed that it was an alternating series,
and i predicted that it converges absolutely.
However, i'm missing steps.

Comparison test

$\displaystyle \frac{n}{\sqrt{1 + n^6}} \:< \:\frac{n}{\sqrt{n^6}} \:=\:\frac{n}{n^3}\:=\:\frac{1}{n^2}$

Hence: .$\displaystyle \sum^{\infty}_{n=1}\frac{n}{\sqrt{1+n^6}} \:< \:\sum^{\infty}_{n=1}\frac{1}{n^2}$ . . . a convergent $\displaystyle p$-series

Therefore: .$\displaystyle \sum^{\infty}_{n=0}\frac{n}{\sqrt{1+n^6}}$ converges.

• Jan 30th 2007, 08:47 PM
rcmango
this is converging to 0 correct?

also, by showing the steps you've provided, would this be enough satisfactory work for proof of this answer.

thanks.