Wallis product

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In mathematics, the Wallis product for π, published in 1656 by John Wallis, states that Wallis product_sentence_0

Proof using integration Wallis product_section_0

(This is a form of Wallis' integrals.) Wallis product_sentence_1

Integrate by parts: Wallis product_sentence_2

This result will be used below: Wallis product_sentence_3

Repeating the process, Wallis product_sentence_4

Repeating the process, Wallis product_sentence_5

By the squeeze theorem, Wallis product_sentence_6

Proof using Euler's infinite product for the sine function Wallis product_section_1

While the proof above is typically featured in modern calculus textbooks, the Wallis product is, in retrospect, an easy corollary of the later Euler infinite product for the sine function. Wallis product_sentence_7

Relation to Stirling's approximation Wallis product_section_2

Derivative of the Riemann zeta function at zero Wallis product_section_3

The Riemann zeta function and the Dirichlet eta function can be defined: Wallis product_sentence_8

Applying an Euler transform to the latter series, the following is obtained: Wallis product_sentence_9

See also Wallis product_section_4

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Wallis product.