Ramanujan’s Series for 1∕π

Shaun Cooper
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Shaun Cooper: Massey University, Institute of Natural and Mathematical Science

Chapter Chapter 14 in Ramanujan's Theta Functions, 2017, pp 613-666 from Springer

Abstract: Abstract We present a variety of techniques for analyzing some remarkable series for 1∕π, such as ∑ n = 0 ∞ 2 n n 3 n + 5 42 1 4096 n = 8 21 × 1 π , $$\sum _{n=0}^{\infty }{2n\choose n}^{3}\left (n + \frac{5} {42}\right )\left ( \frac{1} {4096}\right )^{n} = \frac{8} {21} \times \frac{1} {\pi },$$ that were first studied by Ramanujan. It is shown how to classify the series by level and degree. We also obtain iterative processes that converge rapidly to 1∕π.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-56172-1_15

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DOI: 10.1007/978-3-319-56172-1_15

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