I noticed that a lot of the time, people ask whether the following sum converges:
$$\sum_{n=1}^\infty\frac{\sin(n)}n$$
Though I've never stopped to ask what it equaled. According to this other post, the sum is given as
$$\sum_{n=1}^\infty\frac{\sin(n)}n=\frac{\pi-1}2$$
The solution involves realizing $\sin(n)=\Im e^{in}$ and the Taylor expansion for the natural logarithm.
While thats great and all, how can I prove this using only real numbers?