What are some examples of non-trivial series that are equal to $0$?

Question

By non-trivial, I mean not only non-zero but also a neat looking one, so something you get by normalizing well known series wouldn't work, example : $\sum_{n=0}^{\infty}\left( \frac{1}{en!} - \frac{1}{2^{n+1}}\right)$.

Answer 1

$$\sum_{k=0}^\infty(-1)^k\frac{\pi^{2k+1}}{(2k+1)!}.$$

Answer 2

$$ \sum_{n=1}^\infty \sin \left(\frac{\pi (2n+1)}{(n+1)n}\right) \sin\left(\frac{\pi}{(n+1)n}\right) $$

$$ \sum_{n=1}^\infty \frac{\left(\cos(\pi n/8) + \cos(7 \pi n/8) - 2 \cos(3 \pi n/8)\right)}{n^2} $$

Answer 3

$$ \sum_{k=0}^{\infty}\binom{1/2}{k}(-1)^k=1-\frac{1}{2}-\frac{1}{8}-\frac{1}{16}-\frac{5}{128}-\frac{7}{256}-\dots=0 $$ (see Binomial series)