# How can you tell if a series is absolutely convergent or conditionally convergent?

• May 1st 2010, 07:43 PM
s3a
How can you tell if a series is absolutely convergent or conditionally convergent?
How can you tell if a series is absolutely convergent or conditionally convergent? I am only able to tell if it's convergent and that's it. I read my book and noticed that if you rearrange terms with negative signs, then the sum could change but I don't see how that works in practice so it would be great if someone can compare and contrast the two cases for me.

Any input would be greatly appreciated!
A series, $\displaystyle \sum a_n$ is "absolutely convergent" if $\displaystyle \sum|a_n|$ converges and "conditionally convergent" if $\displaystyle \sum a_n$ converges but $\displaystyle \sum|a_n|$ does not.
Obviously, any series consisting of only positive numbers has $\displaystyle |a_n|= a_n$ and so converges absolutely if and only if it converges. Since, if a series consists only of negative numbers, we could factor out "-1" and have a series of positive numbers, so, similarly, a sequence of negative numbers is absolutely convergent if and only if it converge.
The simplest example of a conditionally convergent sequence is $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}$. That converges by the "alternating sequence test" but $\displaystyle \sum_{n=1}^\infty \left|\frac{(1)^n}{n}= \sum_{n=1}^\infty \frac{1}{n}$ does not converge by the integral test.