A242616 Decimal expansion of lim_(n->infinity) ((Sum_(k=1..n) 1/sqrt(k)) - (Integral_{x=1..n} 1/sqrt(x))), a generalized Euler constant which evaluates to zeta(1/2) + 2.
5, 3, 9, 6, 4, 5, 4, 9, 1, 1, 9, 0, 4, 1, 3, 1, 8, 7, 1, 1, 0, 5, 0, 0, 8, 4, 7, 4, 8, 4, 7, 0, 1, 9, 8, 7, 5, 3, 2, 7, 7, 0, 6, 6, 8, 9, 8, 7, 4, 1, 8, 5, 0, 9, 4, 5, 7, 1, 1, 3, 9, 1, 2, 1, 7, 4, 4, 6, 9, 4, 7, 0, 5, 2, 5, 4, 9, 9, 3, 7, 4, 7, 2, 3, 5, 8, 0, 6, 2, 4, 5, 3, 6, 6, 4, 3, 1, 8, 0, 4
Offset: 0
Examples
0.53964549119041318711050084748470198753277...
References
- Vasile Berinde and Eugen Păltănea, Gazeta Matematică - A Bridge Over Three Centuries, Romanian Mathematical Society, 2004, pp. 113-114.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.5.3, p. 32.
- A. G. Ioachimescu, Problem 16, Gazeta Matematică, Vol. 1, No. 2 (1895), p. 39.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Chao-Ping Chen, Ioachimescu's constant, Research Group in Mathematical Inequalities and Applications, Vol. 13. No. 1 (2010).
- Alina Sîntămărian, A Generalisation of Ioachimescu's Constant, The Mathematical Gazette, Vol. 93, No. 528 (2009), pp. 456-467.
- Alina Sîntămărian, Regarding a generalisation of Ioachimescu's constant, The Mathematical Gazette, Vol. 94, No. 530 (2010), pp. 270-283.
- Alina Sîntămărian, Sequences that converge quickly to a generalized Euler constant, Mathematical and Computer Modelling, Vol. 53, No. 5-6 (2011), pp. 624-630.
- Xu You, Di-Rong Chen, and Hong Shi, Some new sequences that converge to the Ioachimescu constant, Journal of Inequalities and Applications, Vol. 2016, No. 1 (2016), Article 148.
Programs
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Magma
SetDefaultRealField(RealField(100)); L:=RiemannZeta(); 2 + Evaluate(L, 1/2) // G. C. Greubel, Sep 04 2018
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Mathematica
RealDigits[Zeta[1/2] + 2, 10, 100] // First
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PARI
default(realprecision, 100); zeta(1/2)+2 \\ G. C. Greubel, Sep 04 2018
Formula
Equals zeta(1/2) + 2.
Comments