# Absolute convergence of series?

1. Nov 12, 2012

### SMA_01

The question is: Show that if $\sum$an from n=1 to ∞ converges absolutely, then $\sum$an2 from n=1 to converges absolutely.

I'm not sure which approach to take with this.

I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or positive. But i'm not sure where to go with this.

Any help is appreciated.

Thank you.

2. Nov 12, 2012

### micromass

Staff Emeritus
You know that if $a_n$ is small (=smaller than 1), then $a_n^2$ is even smaller. So the squared series is smaller than the original series.

Can you do something with this idea??

3. Nov 13, 2012

### SMA_01

So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?

4. Nov 13, 2012

### micromass

Staff Emeritus
Yes, if it's greater than 1 then there is a problem. Try to find a way to solve this.

5. Nov 13, 2012

### SMA_01

Do you think I can use what I stated in my first post?

6. Nov 13, 2012

### micromass

Staff Emeritus
Uh, not really. Well,it's not wrong what you said, but it won't help you much further.