A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.

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

Offset

0,1

Comments

In 1697, Johann Bernoulli explores this curve and finds its minimum and the area under the curve from 0 to 1, all this without the benefit of the exponential function.

References

Jonathan Borwein, David H. Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, 2004, p. 44.

William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ, 2005, pp. 46-51.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.

Paul J. Nahin, Inside Interesting Integrals, 2nd ed., Springer, New York, 2020, p. 229.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Johann Bernoulli, Demonstratio methodi analyticæ, qua determinata est aliqua quadratura exponentialis per Seriem, Opera omnia, Vol. 3 (1697), pp. 376-381.

M. L. Glasser, A note on Beukers's and related double integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

V. Yu. Irkhin, Sophomore's dream function: asymptotics, complex plane behavior and relation to the error function, arXiv:2501.10936 [math.CA], 2025-2026.

M. S. Klamkin, Problem E 2216, The American Mathematical Monthly, Vol. 77, No. 2 (1970), p. 192; A Comparison of Integrals, Solution to Problem E 2216 by R. A. Groeneveld, ibid., Vol. 77, No. 10 (1970), p. 1114; Comment by C. D. Olds, ibid., Vol. 78, No. 6 (1971), pp. 675-676.

Gábor Román, Extension of the Sophomore’s Dream, Analele ştiinţifice ale Universităţii "Ovidius" Constanţa, Seria Matematică, Vol. 29, No. 1 (2021), pp. 211-218.

Alina Sîntămărian and Ovidiu Furdui, Power Series, in: Sharpening Mathematical Analysis Skills, Problem Books in Mathematics, Springer, Cham, 2021, pp. 90-91.

Eric Weisstein's World of Mathematics, Power Tower.

Eric Weisstein's World of Mathematics, Sophomore's Dream.

Wikipedia, Sophomore's dream.

Formula

Equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012

From Petros Hadjicostas, Jun 29 2020: (Start)

Equals -Integral_{x=0..1, y=0..1} (x*y)^(x*y)/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to Integral_{x=0..1} x^x dx.)

Equals -Integral_{x=0..1} x^x*log(x) dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.)

Without using the results in Glasser (2019), notice that Integral x^x*(1 + log(x)) dx = x^x + c, which implies Integral_{x=0..1} x^x dx = -Integral_{x=0..1} x^x*log(x) dx. (End)

Example

0.78343051071213440705926438652697546940768199014693095825541782270...

Mathematica

RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* Robert G. Wilson v, Jan 31 2005 *)

Prog

(PARI) -sumalt(n=1, (-1/n)^(n)) \\ Michel Marcus, Oct 15 2015
(SageMath) numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # G. C. Greubel, Mar 01 2019

Crossrefs

Cf. A137420 (continued fraction expansion).

Cf. A073009. The minimum point on the curve x^x is (A068985, A072364).

Sequence in context: A020843 A241296 A388556 * A133613 A388808 A379214

Adjacent sequences: A083645 A083646 A083647 * A083649 A083650 A083651

Keyword

cons,nonn

Author

Eric W. Weisstein, May 01 2003

Status

approved