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Proof of Convergence

  1. Dec 9, 2014 #1
    1. The problem statement, all variables and given/known data

    I have been asked to prove the convergence or otherwise of ∑n=1 n/(3n + n2).

    In the example solution, with the aim to prove divergence by comparison with the Harmonic Series, the lecturer has stated that n/(3n + n2) ≥ n/(4n2) = 1/4n and which diverges to +∞.

    I was wondering how to arrive at the decision to write n/(3n + n2) ≥ n/(4n2) came from, and how I would arrive at a similar inequality in further examples.
     
  2. jcsd
  3. Dec 9, 2014 #2

    Ray Vickson

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    For ##a,b,c,> 0## you have ##a/b > a/c## if ##b < c##. In other words, if you make the denominator bigger you make the fraction smaller. Of course, in this case you have ##3n + n^2 < 4 n^2## for ##n > 1## (when ##n < n^2##, so ##3n < 3 n^2##).
     
  4. Dec 9, 2014 #3
    I see, thank you very much.
     
  5. Dec 10, 2014 #4

    statdad

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    A little simpler:
    [tex]
    \dfrac n{n^2+3n} = \dfrac{n}{n(n+3)} = \dfrac{1}{n+3} \ge \dfrac 1 {2n}
    [/tex]
    for [itex] n \ge 3 [/itex]
     
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