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Vanishing infinite series involving hyperbolic cosine

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Posts: 20
Location: Israel
hi, here is a cool series which sums to 0:

$$\large \sum_{n=0}^{\infty} \frac{(-1)^{ \frac{n(n+1)}{2} }\mathbb{e}^{(-1)^{n+1} \frac{\pi}{4} (2n+1)} }{ \cosh \left( \frac{\pi}{2}\, (2n+1)\right)}=0$$

How would you go about proving that?

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