calculus - Is $sum sin^2(k)/k$ Convergent?

A student recently used the series $\displaystyle\sum_{k=1}^\infty\frac{\sin^2k}{k}$ as an example of a divergent series whose terms tend to $0$. However, I'm having trouble convincing myself that this series does in fact converge. Anyone have any ideas?

Answer

The series diverges. Notice

$$\begin{align} \sin^2(k) + \sin^2(k+1) &= \frac12(1-\cos(2k)) + \frac12(1-\cos(2k+2))\\ &= 1 - \cos(1)\cos(2k+1)\\ &\ge 1 - \cos(1)\end{align}$$

We have

$$\sum_{k=1}^{2N} \frac{\sin^2(k)}{k} = \sum_{k=1}^N\left(\frac{\sin^2(2k-1)}{2k-1} + \frac{\sin^2(2k)}{2k}\right) \ge \frac{1-\cos(1)}{2}\sum_{k=1}^N\frac{1}{k}$$
which diverges to $\infty$ as $N \to \infty$.