A245637 -id:A245637 - cp's OEIS frontend

A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

			1.291285997062663540407282590595600541498619368...

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

Kenny Lau, Table of n, a(n) for n = 1..10001
Johan Bernoulli, Demonstratio Methodi Analyticae, qua usus est pro determinanda aliqua Quadratura exponentiali per seriem, Actis Eruditorum A (1697), p. 131. Collected in Opera Omnia, vol. 3, 1742. See p. 376ff.
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Jaroslav Hančl and Simon Kristensen, Metrical irrationality results related to values of the Riemann zeta-function, arXiv:1802.03946 [math.NT], 2018.
Randall Munroe, Approximations, xkcd Web Comic #1047, Apr 25 2012.
Simon Plouffe, Sum(1/n^n, n=1..infinity). [internet archive]
Eric Weisstein's World of Mathematics, Power Tower.
Eric Weisstein's World of Mathematics, Sophomore's Dream.

Cf. A077178 (continued fraction expansion).

Cf. A083648, A229191, A245637.

evalf(Sum(1/n^n, n=1..infinity), 120); # Vaclav Kotesovec, Jun 24 2016

RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]]

suminf(n=1,n^-n) \\ Charles R Greathouse IV, Apr 25 2012

Equals Integral_{x = 0..1} dx/x^x.
Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012
Approximately log(3)^e, see Munroe link. - Charles R Greathouse IV, Apr 25 2012
Another approximation is A + A^(-19), where A is Glaisher-Kinkelin constant (A074962). - Noam Shalev, Jan 16 2015
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals -Integral_{x=0..1, y=0..1} dx dy/((x*y)^(x*y)*log(x*y)). (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the integral Integral_{x = 0..1} dx/x^x.)
Equals -Integral_{x=0..1} log(x)/x^x dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.) (End)

A359283 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^2) dx.

Original entry on oeis.org

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

Offset: 0

Peter Bala, Dec 24 2022

For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.

			0.46230371153732107718203962858827744096102603704840...

Peter Bala, Borel summation of a family of divergent series
Wikipedia, Sophomore's Dream

Cf. A245637, A253299, A359282, A359284, A359285, A359286.

evalf(int(1/x^(x^2), x = 1..infinity), 100);

NIntegrate[1/x^(x^2), {x, 1, Infinity}, WorkingPrecision -> 105] // RealDigits // First

Equals Integral_{x = 1..oo} 1/(2*x - 1)^x dx.

Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(2*n + 1)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1)^(n+1)/(2*n - 1)^n = Integral_{x = 0..1} x^(x^2) dx. See A253299.

A359284 Decimal expansion of Integral_{x = 0..1} 1/x^(x^3) dx.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Dec 24 2022

			1.06551820592764917586382140548454723153980227909982...

Wikipedia, Sophomore's Dream

Cf. A245637, A253299, A359282, A359283, A359285, A359286.

Maple
```
evalf(int(1/x^(x^3), x = 0..1), 110);
```

NIntegrate[1/x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First

PARI
```
intnum(x = 0, 1, x^(-x^3))
```

Equals Sum_{n >= 1} 1/(3*n - 2)^n.

More generally, Integral_{x = 0..1} 1/x^(t*x^3) dx = Sum_{n >= 1} t^(n-1)/(3*n - 2)^n. See A359285 (case t = -1).

A359285 Decimal expansion of Integral_{x = 0..1} x^(x^3) dx.

Original entry on oeis.org

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

Offset: 0

Peter Bala, Dec 24 2022

			0.94031808668190698289736564174297672581175110149306...

Wikipedia, Sophomore's Dream

Cf. A245637, A253299, A359282, A359283, A359284, A359286.

Maple
```
evalf(int(x^(x^3), x = 0..1), 110);
```

NIntegrate[x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First

PARI
```
intnum(x = 0, 1, x^(x^3))
```

Equals Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n.

A359286 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^3) dx.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Dec 24 2022

For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.

			0.35854271600033996570705760779181131168203620572130...

Peter Bala, Borel summation of a family of divergent series
Wikipedia, Sophomore's Dream

Cf. A245637, A253299, A359282, A359283, A359284, A359285.

evalf(int(1/x^(x^3), x = 1..infinity), 110);

NIntegrate[1/x^(x^3), {x, 1, Infinity}, WorkingPrecision -> 110] // RealDigits // First

Equals Integral_{x = 1..oo} 1/(3*x - 2)^x dx.

Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(3*n + 2)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n = Integral_{x = 0..1} x^(x^3) dx. See A359285.

A229191 Decimal expansion of the integral_{x=0..Infinity} 1/x^x dx.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 15 2013

"The function x^x grows even more quickly than Gamma(x) and the integral {0-inf} 1/x^x dx = 1.9954559575... and the integral {1-inf} 1/x^x dx = 0.7041699604... ." [Finch]

			1.9954559575001380004187246984527243520862166369679788727883...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 263.

Cf. A058655, A073009, A245637.

evalf(int(1/x^x, x=0..infinity), 120);  # Alois P. Heinz, Dec 30 2021

RealDigits[ NIntegrate[ 1/x^x, {x, 0, 100}, MaxRecursion -> 5000, MaxPoints -> 5000, AccuracyGoal-> 111, PrecisionGoal -> 111, WorkingPrecision -> 120], 10, 111][[1]]

cp's OEIS Frontend

A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A359283 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^2) dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A359284 Decimal expansion of Integral_{x = 0..1} 1/x^(x^3) dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A359285 Decimal expansion of Integral_{x = 0..1} x^(x^3) dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A359286 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^3) dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A229191 Decimal expansion of the integral_{x=0..Infinity} 1/x^x dx.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica