Results 1 to 3 of 3

Math Help - Apostol Section 10.20 #51

  1. #1
    Junior Member
    Joined
    Sep 2011
    Posts
    70
    Thanks
    2

    Apostol Section 10.20 #51

    Given a covergent series \textstyle \sum a_n, where each a_n \ge 0, prove that \textstyle \sum \sqrt{a_n} n^{-p} converges if p>\frac 1 2.

    My attempt so far:
    I want to say that, if \textsyle \sum a_n is convergent, then for some N, we have that for all n>N, 0\le a_n\le \frac 1 n. The conclusion follows quickly after that, but I do not think that this original statement is necessarily true. For instance, consider

    \sum a_n where
    a_n = \left\{ \begin{array}{lr} \frac 1 n & : n\text{ is a square}\\  \frac 1 {n^2} & : \text{otherwise}\end{array} \right.

    Edit: Just realized that this doesn't explicitly prove my original inequality to be incorrect, but obviously a slight adjustment to this would.
    Last edited by process91; October 11th 2011 at 05:58 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1

    Re: Apostol Section 10.20 #51

    Hi process91,

    I think you are correct to doubt the validity of saying a_n \leq 1/n... it's not necessarily so.

    You might consider applying the Cauchy-Schwartz inequality to \sum \sqrt{a_n} n^{-p}. Maybe you can bound it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2011
    Posts
    70
    Thanks
    2

    Re: Apostol Section 10.20 #51

    Yup, that got it.

    0\le\left(\sum_{n=1}^k \sqrt{a_n} n^{-p} \right) ^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 \sum a_n and \sum n^{-2p}, which exist since these both converge.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: March 10th 2013, 05:13 PM
  2. apostol calculus, trouble simplifying integral
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 12th 2010, 01:09 PM
  3. Proof of Green's theorem in Apostol's book
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: April 26th 2010, 10:06 PM
  4. apostol and archimedes
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 21st 2009, 09:34 PM
  5. Apostol - Analytic Number Theory
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: February 13th 2009, 10:01 AM

/mathhelpforum @mathhelpforum