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.
 ^2 \le \sum_{n=1}^k a_n \sum_{n=1}^k n^{-2p} \le MN)
where M and N are the bounds of the partial summations for and 
, which exist since these both converge.
