Q:

Convergence of this series $\sum\limits_{n=1}^{\infty} \ln\frac{\cosh\frac{1}{n}}{\cos\frac{1}{n}}$

I'd like to know convergence of  this series 

$$\sum\limits_{n=1}^{\infty} \ln\frac{\cosh\frac{1}{n}}{\cos\frac{1}{n}}$$

but  I can not guess what test can be useful. I wounder if anybody here could give me hints? 
Thanks in advance.  

A:

By expanding the functions
$$
\ln \frac{\cosh {\frac 1n}}{\cos \frac 1n} = \ln \frac{1 + \frac{1}{2n^2} + o\left(\frac 1{n^2}\right)}{1 - \frac{1}{2n^2} + + o\left(\frac 1{n^2}\right)} = \ln \left({1 + \frac{1}{n^2} + o\left(\frac 1{n^2}\right)}\right) = \frac {1}{n^2} + o\left(\frac 1{n^2}\right),
$$
the series is easily seen as convergent.

