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Optimization Doodles and Dave's Amazing Summation (Redo)

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Dave has an amazing summation: n=1n13e2πn1=124\displaystyle \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n} - 1} = \frac{1}{24}

Moreover, this is exactly equal to the integral!

0x13e2πx1dx=n=1n13e2πn1=124\displaystyle \int_0^\infty \frac{x^{13}}{e^{2 \pi x} - 1} dx = \displaystyle \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n} - 1} = \frac{1}{24}

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Published: Nov 4, 2022

Last Modified: Apr 28, 2023

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