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 : $sum_{n=0}^{infty}left( frac{1}{en!} - frac{1}{2^{n+1}}right)$.

Sil · Answer

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

Robert Israel · Answer

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

Yves Daoust · Answer

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

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

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