Convergence tests for series

**Dragonkiller** · August 24th 2012, 04:09 AM

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

**Plato** · August 24th 2012, 05:11 AM

Originally Posted by Dragonkiller

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

Now that answers both questions, but how?

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