A349522 Decimal expansion of Sum_{k>=2} 1/(k*log(k))^2.

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

Offset: 0

Author

Bernard Schott, Nov 20 2021

Keywords

• nonn
• cons

Comments

Theorem: Bertrand series Sum_{n>=2} 1/(n^q*log(n)^r) is convergent if q > 1.

Application for q = 2 with A201994 (r=-2), A073002 (r=-1), A013661 (r=0), A168218 (r=1), this sequence (r=2).

Examples

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Links

• Wikipédia, Série de Bertrand (in French).
• Wikipedia, Joseph Bertrand.

Crossrefs

Cf. A013661, A073002, A168218, A201994.

Programs

PARI

sumpos(k=2, 1/(k*log(k))^2) \\ Michel Marcus, Nov 21 2021

Formula

Equals Sum_{k>=2} 1/(k*log(k))^2.

Equals Integral_{x>=2, y>=2} (zeta(x + y - 2) - 1) dx dy. - Amiram Eldar, Nov 21 2021