Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Divergent Harmonic Series, Convergent P-Series (Cauchy sequences)

  1. Mar 25, 2008 #1
    1. The problem statement, all variables and given/known data
    (a) Show that [itex]\sum \frac 1n[/itex] is not convergent by showing that the partial sums are not a Cauchy sequence
    (b) Show that [itex]\sum \frac 1{n^2}[/itex] is convergent by showing that the partial sums form a Cauchy sequence


    2. Relevant equations
    Given epsilon>0, a sequence is Cauchy if there exists an N such that [itex]|a_m-a_n|<\epsilon[/itex] for every m,n>N.


    3. The attempt at a solution
    For part (a), the sequence terms are [itex]a_n=1+1/2+\ldots+1/n[/itex], so assuming that m>n,
    [tex]|a_m-a_n|=1/m+1/(m-1)+\ldots+1/(n+1)<\frac{m-n}{n+1}<\frac{m-n}{n}[/tex].

    Now, if I take epsilon=1/2 and suppose that m>n>N implies that the distance between two elements is less than epsilon. But m=2n>n>N gives the difference to be 1, which is greater than epsilon, so the sequence is not Cauchy. I think I've gotten this half - am I correct??

    For part (b), I'm not sure how to do it. The sequence terms are [itex]a_n=1+1/2^2+\ldots+1/n^2[/itex], so again assuming that m>n, we have

    [tex]|a_m-a_n|=1/m^2+1/(m-1)^2+\ldots+1/(n+1)^2<\frac{m-n}{(n+1)^2}<\frac{m-n}{n^2}<\frac{m}{n^2}[/tex]

    I want to be able to find N such that m>n>N implies that this difference is less than epsilon, right? To do that, since I've assumed that m>n, if I can eliminate m from the expression, I'm good to go, but I can't figure out how to do it. Help!
     
    Last edited: Mar 25, 2008
  2. jcsd
  3. Mar 26, 2008 #2
    I'm grasping at straws here, but I want to show that [itex]m/n^2<\epsilon[/itex]. So I can say, since [itex]n<m[/itex], that [itex]n/n^2=1/n<m/n^2<\epsilon[/itex].

    I know this probably isn't right, but if anybody could give me a hand it'd be appreciated.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook