For $f\left(x\right)=\sum _{n=1}^{\infty }\:\frac{\cos^{2n}\left(x\right)}{n}$, show that $f\left(\frac{\pi }{6}\right)\:=\:ln\left(4\right)$
Solution 1:
Recall that $\ln(1+x)=\sum_{n=1}^{\infty}\frac{x^{n}}{n}(-1)^{n+1}$ for $|x|<1$. Put $x=-\frac{3}{4}$, then we have $\sum_{n=1}^{\infty}\frac{(3/4)^{n}}{n}=-\ln(1-\frac{3}{4})=\ln4$.