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

1. 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!

2. The most general method is to use the definition.

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.

Note that many of the basic "convergence" tests require positive values and so really test for absolute convergence. That's why we always take the absolute value of a power series to test for convergence and why a power series converges absolutely inside it radius of convergence.

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.

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