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 :)

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?