Alternating Series

**BACONATOR** · July 18th 2008, 09:59 AM

I have trouble with alternating series.
I have to test the series for convergence or divergence.
Any help would be appreciated
Thanks!

**Moo** · July 18th 2008, 11:18 AM

Originally Posted by BACONATOR

I have trouble with alternating series.
I have to test the series for convergence or divergence.
Any help would be appreciated
Thanks!

Hello,

Remember that for an alternating series to converge $\sum_{n} (-1)^n \ a_n$ , you have to check that :

- $\lim_{n \to \infty} a_n=0$
- $a_n$ is a decreasing sequence

But here, one can see that when n approaches high values, +2 can be neglected.

Thus the general term is equivalent to $(-1)^n \frac{n}{\sqrt{n^3+2}} \sim (-1)^n \frac{n}{\sqrt{n^3}}=(-1)^n \frac{n}{n^{3/2}}=(-1)^n \frac{1}{n^{1/2}}$ , which is an alternating Riemann series that converges.

**flyingsquirrel** · July 18th 2008, 11:53 AM

Hi

Originally Posted by Moo

Thus the general term is equivalent to $(-1)^n \frac{n}{\sqrt{n^3+2}} \sim (-1)^n \frac{n}{\sqrt{n^3}}=(-1)^n \frac{n}{n^{3/2}}=(-1)^n \frac{1}{n^{1/2}}$ , which is an alternating Riemann series that converges.

You can't use approximations ( $\sim$ ) in this case since the sign of $(-1)^n\frac{n}{\sqrt{n^3+2}}$ changes : The theorem you're using states that if ${\color{red}a_n,\,b_n>0}$ , if $a_n\sim b_n$ and if $\sum a_n$ converges then $\sum b_n$ converges.

**Mathstud28** · July 18th 2008, 01:07 PM

Originally Posted by BACONATOR

I have trouble with alternating series.
I have to test the series for convergence or divergence.
Any help would be appreciated
Thanks!

We can see that this satisfies the two conditions of the converging series

$\sum_{n=0}^{\infty}(-1)^na_n$

Firstly

$\exists{N}\backepsilon\forall{n>N}\quad{a_{n+1}<a_ n}$

Furthermore

$\lim_{n\to\infty}a_n=0$

$\Rightarrow\quad\sum_{n=0}^{\infty}(-1)^n\frac{n}{\sqrt{n^3+2}}~~\text{converges}$

Thread: Alternating Series

LinkBack

Thread Tools

Display

Alternating Series

Similar Math Help Forum Discussions

alternating series help

alternating series

Alternating series

alternating series

Alternating Series

Search Tags