Apostol Section 10.20 #51

Given a covergent series , where each , prove that converges if .

My attempt so far:

I want to say that, if is convergent, then for some N, we have that for all n>N, . The conclusion follows quickly after that, but I do not think that this original statement is necessarily true. For instance, consider

where

Edit: Just realized that this doesn't explicitly prove my original inequality to be incorrect, but obviously a slight adjustment to this would.

Re: Apostol Section 10.20 #51

Hi process91,

I think you are correct to doubt the validity of saying ... it's not necessarily so.

You might consider applying the Cauchy-Schwartz inequality to . Maybe you can bound it.

Re: Apostol Section 10.20 #51

Yup, that got it.

where M and N are the bounds of the partial summations for and , which exist since these both converge.