A354014 Decimal expansion of Sum_{n>0} u(n) where u(n) is the unique positive solution to the equation Integral_{u(n)..1} e^t/t dt = n.

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

Offset: 1

Author

Bernard Schott, May 14 2022

Keywords

• nonn
• cons

Comments

Near infinity, u(n) ~ e^(lambda)/e^n, with lambda = A229837 = Integral_{t=0..1} (e^t-1)/t dt, so this series Sum_{n>0} u(n) is convergent.

Examples

			1.24900773295782056784977184983154145525925969937566...
		

References

• Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.30 pp. 252 and 450-451.

Links

• Amiram Eldar, Plot of u(n) for n=1..10
• Amiram Eldar, Logarithmic plot of u(n) for n=1..10
• Amiram Eldar, Plot of Sum_{i=1..n} u(i) for n=1..10

Crossrefs

Cf. A229837.

Programs

PARI

N = 100;
default(realprecision, N);
u(n) = {my(integ = intformal(sum(k=1, N, x^(k-1)/k!), x)); solve(y=1./10^N, 1, subst(integ, x, 1) - log(y) - subst(integ, x, y) - n);}
sum(k=1, N, u(k)) \\ Michel Marcus, May 20 2022

Extensions

More terms from Amiram Eldar, May 14 2022