From Fourier series to infinite product representations of $\pi$ and infinite-series forms for its positive powers

Jean-Christophe Pain

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arXiv · 2210.09123

From Fourier series to infinite product representations of $\pi$ and infinite-series forms for its positive powers

Jean-Christophe Pain

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Abstract

In this article, we derive, using Fourier series and multiple derivative of the function $\pi/\sin(\pi x)$, series representations for positive powers of $\pi$. We also show that the Euler-Wallis product can be easily obtained from the same formalism and deduce infinite products representations of $\pi$.

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Jean-Christophe Pain. 2022-08-25. From Fourier series to infinite product representations of $\pi$ and infinite-series forms for its positive powers. https://arxiv.org/abs/2210.09123

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