Series with Hyperbolic and Trigonometric functions

  • Thread starter azatkgz
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  • #1
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Homework Statement


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]






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
 

Answers and Replies

  • #2
Gib Z
Homework Helper
3,346
6
Sorry its not immediately obvious to me how you got the your first line of working to your second.
 

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