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Proving summation series inequality

  1. Oct 28, 2013 #1
    Question
    http://puu.sh/52zAa.png [Broken]

    Attempt
    http://puu.sh/52AVq.png [Broken]

    I've attempted to use Riemann sums and use the integral to prove the inequality, not sure if this was the right approach to start with as I am now stuck and don't see what to do next.

    For part (b), I know that if (2√n -2) → ∞ as n → ∞, then Sn → ∞ for n → ∞ hence the summation series is divergent.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 28, 2013 #2

    mfb

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    Staff: Mentor

    ##S_n > 2 \sqrt{n}-2## and ##S_n > \sqrt{n}## are not the same.
    I think you are supposed to use induction in (a). The integral approach works, but it needs more mathematics.

    Good, as (2√n -2) → ∞ for n → ∞ is true.
     
  4. Oct 29, 2013 #3
    does it help to shift the series from k=1 to k=2 which differ only by 1? if you can prove the inequality holds for Sn>2sqrt(n)>sqrt(n) since n>0

    maybe there are cases where you can shift the sum by a real number so that the first term of the sum is equal to k in the integral 2sqrt(n)-k ? Euler proved the series convergence for k=1 to n=infinity, 1/n^2=pi^2/6
     
    Last edited: Oct 29, 2013
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