# Convergence tests for series

• Aug 24th 2012, 04:09 AM
Dragonkiller
Convergence tests for series
Hi I was wondering if anyone could help me with a quick question:
If both $\displaystyle \sum_{n=1}^{\infty }a_{n}$ and $\displaystyle \sum_{n=1}^{\infty }|a_{n}|$ converge and both sums are equal what can you conclude about the two series?
If one of $\displaystyle \sum_{n=1}^{\infty }a_{n}$ and $\displaystyle \sum_{n=1}^{\infty }|a_{n}|$ converges and the other one diverges, which converges and which diverges?
Thanks :)
• Aug 24th 2012, 05:11 AM
Plato
Re: Convergence tests for series
Quote:

Originally Posted by Dragonkiller
Hi I was wondering if anyone could help me with a quick question:
If both $\displaystyle \sum_{n=1}^{\infty }a_{n}$ and $\displaystyle \sum_{n=1}^{\infty }|a_{n}|$ converge and both sums are equal what can you conclude about the two series?
If one of $\displaystyle \sum_{n=1}^{\infty }a_{n}$ and $\displaystyle \sum_{n=1}^{\infty }|a_{n}|$ converges and the other one diverges, which converges and which diverges?

There is a standard theorem that says: If a series converges absolutely then it converges conditionally.
i.e. If $\displaystyle \sum_{n=1}^{\infty }|a_{n}|$ converges then $\displaystyle \sum_{n=1}^{\infty }a_{n}$ converges.

Now that answers both questions, but how?