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Some series for π

Published online by Cambridge University Press:  31 October 2008

C. E. Walsh
C. E. Walsh
Affiliation:
74 Serpentine Avenue, Ballsbridge, Dublin
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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).

Type
Research Article
Information
Edinburgh Mathematical Notes , Volume 37 , January 1949 , pp. 20 - 21
Copyright
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