Series with Hyperbolic and Trigonometric functions

[azatkgz]

Homework Statement

Determine whether the series converges and diverges.

$$\sum_{n=3}^{\infty}\ln \left(\frac{\cosh \frac{\pi}{n}}{\cos \frac{\pi}{n}}\right)$$

The Attempt at a Solution

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

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

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

series converges

 

[Gib Z]

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