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

Mathematics Asked by the_firehawk on December 2, 2020

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

3 Answers

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

Answered by Sil on December 2, 2020

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

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

Answered by Robert Israel on December 2, 2020


Answered by Yves Daoust on December 2, 2020

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