Prove that $\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$ converges absolutely.

Question

Prove that $$\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$$ converges absolutely.

The answer here suggests to use the Limit Comparison Test but it works for $a_n \geq 0$ while $\ln(\cos (1/n))<0$. Also the limit given in the answer is $-\frac{1}{2}$ while the test gives results for positive limit values only. That post is $6$ years old so I didn't leave this as a comment.

Answer 1

As $\;0<\cos\frac 1n<1$, we have $\;\Bigl|\log\cos\frac1n\biggr|=\log\biggl(\frac 1{\cos\tfrac1n}\biggr)$.

Now $\cos\frac1n=1-\frac1{2n^2}+o\bigl(\frac1{n^2}\bigr)$, so $$\frac 1{\cos\tfrac1n}=\frac 1{1-\tfrac1{2n^2}+o\bigl(\frac1{n^2}\bigr)}=1+\frac1{2n^2}+o\bigl(\frac1{n^2}\bigr)\sim_\infty1+\frac1{2n^2},$$ and ultimately $$\Bigl|\log\cos\frac1n\biggr|\sim_\infty \log\Bigl(1+\frac1{2n^2}\Bigr)\sim_\infty \frac1{2n^2}, $$ which converges.