The harmonic series $\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$ is divergent.
Since $ \frac1n = \frac{\sin^2\left(n\right)}{n} + \frac{\cos^2\left(n\right)}{n}$. I'm wondering if $\sum_{n=1}^\infty \frac{\sin^2\left(n\right)}{n}$ and $\sum_{n=1}^\infty \frac{\cos^2\left(n\right)}{n}$ are divergent.
Here is my try:
\begin{align*} \cos 2n &= 2 \cos^2 n - 1 \\ &= 1 - 2 \sin^2 n \\ \\ \sum_{n=1}^\infty \frac{\sin^2\left(n\right)}{n} &= \sum_{n=1}^\infty \frac{2\sin^2\left(n\right)}{2n} \\ &= \sum_{n=1}^\infty \frac{1 - \cos 2n}{2n} \\ \\ \sum_{n=1}^\infty \frac{\cos^2\left(n\right)}{n} &= \sum_{n=1}^\infty \frac{2\cos^2\left(n\right)}{2n} \\ &= \sum_{n=1}^\infty \frac{1 + \cos 2n}{2n} \end{align*}
How should I proceed?
