# Does a series converge if it's absolutely convergent?

• Jun 27th 2010, 04:18 PM
Also sprach Zarathustra
Does a series converge if it's absolutely convergent?
A stupid one... (just to be sure)

If I have a series that is converge absolutely so the series is converges?

(it is implies from Cauchy criterion?)

Thanks!
• Jun 27th 2010, 04:33 PM
Prove It
Quote:

Originally Posted by Also sprach Zarathustra
A stupid one... (just to be sure)

If I have a series that is converge absolutely so the series is converges?

(it is implies from Cauchy criterion?)

Thanks!

If a series is absolutely convergent, then it is convergent.

This is easily proven using the comparison test.
• Jun 27th 2010, 04:35 PM
skeeter
• Jun 27th 2010, 05:04 PM
Also sprach Zarathustra
Quote:

Originally Posted by Prove It
If a series is absolutely convergent, then it is convergent.

This is easily proven using the comparison test.

I think you wrong by saying "This is easily proven using the comparison test."

I know what you mean, but it's not so true....

ehmmm...
• Jun 27th 2010, 05:16 PM
Prove It
Actually it is.

Note that for any infinite series $\displaystyle \sum{a}$, it can never be any greater than the sum of its absolute values.

So $\displaystyle \sum{a} \leq \sum{|a|}$.

Now simply apply the comparison test.

Note though that this only half proves the theorem, however you can also bound the series below...

$\displaystyle \sum{(-|a|)} \leq \sum{a}$

$\displaystyle -\sum{|a|} \leq \sum{a}$.

So we can say

$\displaystyle -\sum{|a|} \leq \sum{a} \leq \sum{|a|}$

and since the series is bounded by the sum of the absolute values, that means if the sum of the absolute values converges, then the series is also convergent.