Thread: 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!

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.

Theorem: Absolute Convergence implies Convergence

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...

Actually it is.

Note that for any infinite series , it can never be any greater than the sum of its absolute values.

So .


Now simply apply the comparison test.


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



.



So we can say



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.