Showing that $-\frac{1}{2}=\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}$

Guide:

Verify by computing the fourier series of $f(x) = \frac{x}{\pi}, -\pi < x<\pi$

$$\frac{x}{\pi} = \frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\sin(nx)$$

and choose an appropriate value of $x$.

A direct approach: $$\sum_{n=1}^\infty\frac{(-1)^n\sin n}n=\sum_{n=1}^\infty\frac{(-1)^n\Im (e^{in})}n=-\Im\sum_{n=1}^\infty\frac{(-1)^{n+1}(e^i)^n}n=-\Im\ln(1+e^i)$$ where the Maclaurin series of $\ln(1+z)$ was used. As can be seen by drawing a diagram, $\arg(1+e^i)=\frac12$. This is thus the imaginary part of the logarithm, and thus the original sum is $-\frac12$.

Its is also possible to calculate the series above using the residue theorem from complex analysis. It goes as follows:$$\sum_{i=-\infty}^\infty \text{Res}\left(\pi\csc(\pi z)\cdot\frac {\sin(z)}z,n\right)=-\text{Res}\left(\pi\csc(\pi z)\cdot\frac {\sin(z)}z,0\right)$$

The LHS becomes: $$\sum_{i=-\infty}^\infty\lim_{z\to n}\left[\pi (z-n)\csc(\pi z)\cdot\frac {\sin(z)}z\right]=\sum_{i=-\infty}^\infty(-1)^n \frac {\sin(n)}n=2\sum_{i=1}^\infty(-1)^n \frac {\sin(n)}n$$

The RHS becomes: $$\lim_{z\to 0}\left[\pi z\csc(\pi z)\cdot\frac {\sin(z)}z\right]=1$$ Putting all together (LHS $=-$ RHS) you get: $$\sum_{i=1}^\infty(-1)^n \frac {\sin(n)}n=-\frac 12$$