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Series with Hyperbolic and Trigonometric functions

  1. Nov 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Determine whether the series converges and diverges.

    [tex]\sum_{n=3}^{\infty}\ln \left(\frac{\cosh \frac{\pi}{n}}{\cos \frac{\pi}{n}}\right)[/tex]






    3. The attempt at a solution

    [tex]\sum_{n=3}^{\infty}\ln \left(\frac{1+\frac{\pi^2}{2n^2}+O(\frac{1}{n^4})}{1-\frac{\pi^2}{2n^2}+O(\frac{1}{n^4})}\right)[/tex]

    [tex]=\sum_{n=3}^{\infty}\ln \left(\left(1+\frac{\pi^2}{2n^2}+O(\frac{1}{n^4})\right)\left(1+\frac{\pi^2}{2n^2}+O(\frac{1}{n^4})\right)\right)=\sum_{n=3}^{\infty}\ln \left(1+\frac{\pi^2}{n^2}+O(\frac{1}{n^4})\right)[/tex]

    [tex]=\sum_{n=3}^{\infty}\left(\frac{\pi^2}{n^2}+O(\frac{1}{n^4})\right)[/tex]

    series converges
     
  2. jcsd
  3. Nov 19, 2007 #2

    Gib Z

    User Avatar
    Homework Helper

    Sorry its not immediately obvious to me how you got the your first line of working to your second.
     
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