Z coth(z) = sum of 2^(2n)B (2n) z^(2n)/(2n)!

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Theorem

The following formula holds for $|z| < \pi$: $$z \mathrm{coth}(z) = \displaystyle\sum_{k=0}^{\infty} 2^{2k} B_{2k} \dfrac{z^{2k}}{(2k)!},$$ where $\mathrm{coth}$ denotes hyperbolic cotangent, $B_{2k}$ denotes Bernoulli numbers, and $(2k)!$ denotes factorial.

Proof

References