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Some series for π
Published online by Cambridge University Press: 31 October 2008
Extract
Consider three sequences αn, Dn, kn (n = 1, 2, 3,…), such that Dnan→0 and, for n > 1,
Then Hence, writing Σ for
,
if either series converges. This will be applied to derive various series for π from the two known results1
,
where Nn = 3. 5. ….(2n + 1).
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- Research Article
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- Copyright © Edinburgh Mathematical Society 1949
References
1 Knopp, , Infinite Series, p. 269, Ex. 110 (a), and p. 246, Ex.2.Google Scholar
1 These rearrangements consist in taking the first term of Σ separately and changing n to n + 1.
2 Another such expression was found by Dalzell, D. P., “On 22/7,” Journ. London Math. Soc., 37 (1944), pp. 133–4.CrossRefGoogle Scholar