# Conditionally convergent series

• Sep 28th 2012, 09:16 AM
nvwxgn
Conditionally convergent series
Consider the power series

$\sum \frac{a^{n+1}}{n+1}\, {(-3)}^{n+1}$

The problem is: there is only one value of a for which the series is conditionally convergent. Find it.

For a = 1/3, the series is alternating and convergent. For a = -1/3, it diverges. I can't think of any number for which it becomes conditionally convergent.

Any help would be appreciated.
• Sep 28th 2012, 09:55 AM
TheEmptySet
Re: Conditionally convergent series
Quote:

Originally Posted by nvwxgn
Consider the power series

$\sum \frac{a^{n+1}}{n+1}\, {(-3)}^{n+1}$

The problem is: there is only one value of a for which the series is conditionally convergent. Find it.

For a = 1/3, the series is alternating and convergent. For a = -1/3, it diverges. I can't think of any number for which it becomes conditionally convergent.

Any help would be appreciated.

You have found what you are looking for. What you may be lacking is the definition of conditionally convergent!

A series is conditionally convergent if it is convergent, but not absolutly convergent.

So when $a=\frac{1}{3}$

You get the series

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

That converges by the A.S.T but

$\sum_{n=0}^{\infty} \bigg| \frac{(-1)^{n+1}}{n+1} \bigg| =\sum_{n=0}^{\infty}\frac{1}{n+1}$

This is the tail end of the harmonic series.
• Sep 28th 2012, 11:50 AM
nvwxgn
Re: Conditionally convergent series
Thanks, I get it now.