Absolute convergence of series?

[SMA_01]

The question is: Show that if $\sum$a_n from n=1 to ∞ converges absolutely, then $\sum$a_n² from n=1 to converges absolutely.

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

I am thinking that since Ʃa_n converges absolutely, |a_n| can be either -a_n or a_n and for Ʃa_n², a_n can be either negative or positive. But I'm not sure where to go with this.

Any help is appreciated.

Thank you.

 

[micromass]

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??

 

[SMA_01]

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

 

[micromass]

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

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

 

[SMA_01]

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

 

[micromass]

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

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