A361972 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k*log(k)) - log(log(n)) ).

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

Offset: 0

Author

Bernard Schott, Apr 08 2023

Keywords

• nonn
• cons

Comments

Let u(n) = Sum_{k=2..n} 1/(k*log(k)) - log(log(n)), then (u(n)) is strictly decreasing and lower bounded by -log(log(2)) = A074785, so (u(n)) is convergent, while the series v(n) = Sum_{k=2..n} 1/(k*log(k)) diverges (see Mathematics Stack Exchange link).

Compare with w(n) = Sum_{k=1..n} 1/k - log(n) that converges (A001620), while the harmonic series H(n) = Sum_{k=1..n} 1/k diverges.

Examples

			0.79467864545289940220389796...
		

References

• J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1.18 p. 23, 1988, Classes Préparatoires aux Grandes Ecoles, Ellipses.

Links

• Mathematics Stack Exchange, Infinite series Sum_{n=2..oo} 1/(n*log(n)).

Crossrefs

Cf. A001620, A074785, A241005.

Programs

Maple

limit(sum(1/(k*log(k)), k=2..n) - log(log(n)), n = infinity);

Formula

Limit_{n->oo} 1/(2*log(2)) + 1/(3*log(3)) + ... + 1/(n*log(n)) - log(log(n)).

Equals A241005 - log(log(2)) = A241005 + A074785. - Amiram Eldar, Apr 08 2023