A226317 Decimal expansion of the constant of Theodorus.

1, 8, 6, 0, 0, 2, 5, 0, 7, 9, 2, 2, 1, 1, 9, 0, 3, 0, 7, 1, 8, 0, 6, 9, 5, 9, 1, 5, 7, 1, 7, 1, 4, 3, 3, 2, 4, 6, 6, 6, 5, 2, 4, 1, 2, 1, 5, 2, 3, 4, 5, 1, 4, 9, 3, 0, 4, 9, 1, 9, 9, 5, 0, 3, 5, 9, 8, 3, 4, 2, 7, 2, 3, 3, 9, 9, 9, 2, 1, 3, 2, 0, 5, 6, 8, 8, 3, 8, 7, 5, 6, 4, 9, 9, 6, 1, 4, 4, 9, 5

Offset: 1

Author

Walter Gautschi (wxg(AT)cs.purdue.edu), Robert G. Wilson v, and Jean-François Alcover, Apr 15 2013

Keywords

• nonn
• cons

Comments

The decimal expansion of the Sum {k>=1} 1/(k^(3/2) + k^(1/2)).

This constant was first identified by Professor Philip J. Davis.

This constant is not in Steven R. Finch, Mathematical Constants, Cambridge, 2003, nor is it in the Inverse Symbolic Calculator (originally by Simon Plouffe & the Borwein brothers).

Examples

			1.86002507922119030718069591571714332466652412152345149304919950359788...
		

References

• Philip J. Davis, Spirals: From Theodorus to Chaos, AK Peters, 1993.
• Julian R. Havil, The Irrationals: A Story of the Numbers You Can't Count On, Princeton University Press, Princeton NJ, 2012, page 277.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1024
• David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
• Ewan Brinkman, Robert Corless, and Veselin Jungic, The Theodorus Variation, Maple Transactions, Vol. 1, No. 2 (2021), Article 14500.
• Steven Finch, Constant of Theodorus [broken link. Probably removed from CiteSeerX]
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663.
• Walter Gautschi, Purdue University, The Spiral of Theodorus, Numerical Analysis, and Special Functions.
• Kevin Ryde, Math-PlanePath, Theodorus Spiral.
• Jörg Waldvogel, Analytic Continuation of the Theodorus Spiral, Seminar für Angewandte Mathematik, ETH Zürich, 2008.
• Eric Weisstein's World of Mathematics, Theodorus's Constant.

Crossrefs

Cf. A072895, A105459, A224269.

Programs

Maple

Digits := 102: evalf(sum((k^(3/2) + k^(1/2))^(-1), k=1..infinity));
# Peter Luschny, Feb 28 2022

Mathematica

digits = 100; 2/Sqrt[Pi]*NIntegrate[(-Exp[t^2])*Log[1 - Exp[-t^2]] - 1, {t, 0, Infinity}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First
(* or *)
a = NSum[1/(k^(3/2) + k^(1/2)), {k, 1, Infinity}, AccuracyGoal -> 2^8, PrecisionGoal -> 2^8, WorkingPrecision -> 2^8, NSumTerms -> 2^15]; RealDigits[a, 10, 105][[1]]

PARI

sumpos(k=1,1/sqrt(k)/(1+k)) \\ Charles R Greathouse IV, Aug 29 2013

PARI

sumalt(k=0,zeta(k+3/2)*(-1)^k) \\ Charles R Greathouse IV, Aug 29 2013

Formula

Sum_{k>=1} 1/(k^(3/2) + k^(1/2)).

Equals -(2/sqrt(Pi)) * Integral_{x>=0} (exp(x^2)*log(1-exp(-x^2))+1) dx (Waldvogel, 2008). - Amiram Eldar, Jul 19 2022