Results 1 to 2 of 2

Thread: Convergence- proving bounded

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    13

    Convergence- proving bounded

    Given sequence, a= 1/1^2+1/2^2+1/3^2+.......+1/n^2

    (1) I have to prove for natural numbers n, such that n is greater than or equal to 2,

    a is less than or equal to 1+1/1*2+1/2*3+1/3*4+.....+1/(n-1)*n

    (2) Prove that a is bounded for natural numbers

    I've been given a hint in the question to use the fact that for n greater than or equal to 2, 1/(n-1)n=1/(n-1) -1/n

    Anyone have any ideas? Anything would be very much appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Hello.

    In part one, you can just compare terms directly. The $\displaystyle n$-term in your first series is $\displaystyle \frac{1}{n^2}$, and the $\displaystyle n$-term in your second series is $\displaystyle \frac{1}{(n-1)n}$. Is it true that $\displaystyle \frac{1}{n^2}\leq\frac{1}{(n-1)n}$ (for $\displaystyle n>2$)? If so, then you have $\displaystyle 1+\frac{1}{2^2}+\ldots+\frac{1}{n^2}<1+\frac{1}{1\ cdot 2}+\ldots+\frac{1}{(n-1)n}$, and therefore, the first series is bounded by the second.

    In part two, we can try to sum up the second series to put a bound on the first. Try writing it down using your hint:

    $\displaystyle 1+\frac{1}{1\cdot 2}+\ldots+\frac{1}{(n-1)n}=1+\frac{1}{2}+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\ldots+(\frac{1}{n-2}-\frac{1}{n-1})+(\frac{1}{n-1}-\frac{1}{n})$

    Anything nice happen? What happens when you take $\displaystyle n\to \infty$? Use this to conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Sep 7th 2011, 05:34 PM
  2. Proving a bounded operator
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 29th 2010, 03:59 PM
  3. Bounded and Convergence Proof Question
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 11th 2010, 10:00 AM
  4. Bounded and Convergence Proof Question
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Mar 11th 2010, 07:45 AM
  5. Replies: 3
    Last Post: Dec 10th 2008, 11:32 AM

Search Tags


/mathhelpforum @mathhelpforum