Conditionally convergent series

Consider the power series

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

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.

Re: Conditionally convergent series

Quote:

Originally Posted by

**nvwxgn** Consider the power series

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

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 $\displaystyle a=\frac{1}{3}$

You get the series

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

That converges by the A.S.T but

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

This is the tail end of the harmonic series.

Re: Conditionally convergent series