Is the Series \(\sum_{i=1}^{\infty} \ln(\cos(\frac{1}{n}))\) Convergent?

TTob
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Homework Statement


Check if the following series is convergent.
<br /> \sum^{\infty}_{i=1}l n(cos(\frac{1}{n}))<br />I have tried a lot of different tests without success.
I need some hint.

Thanks

Homework Equations


The Attempt at a Solution

 
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Hi TTob! :smile:

Hint: this is ln of a product
 
Tayor's Theorem
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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