A355921 -id:A355921 - cp's OEIS frontend

A231132 Decimal expansion of sum_(n=2..infinity) (-1)^n*zeta(n)/n^2.

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

Offset: 0

Jean-François Alcover, Nov 04 2013

Let f(k) = sum_(n=2..infinity) (-1)^n*zeta(n)/n^k, then Euler gamma is f(1) and this constant is f(2).

			0.32034114251279383627256109321177871875321147987620323852089693133571334868...

Eric Weisstein's MathWorld, Euler-Mascheroni Constant

Cf. A001620, A352619, A355921, A365959.

RealDigits[ EulerGamma + Integrate[ LogGamma[x+1]/x, {x, 0, 1}] // N[#, 100]&, 10, 100] // First

sumalt(k=2,(-1)^k*zeta(k)/k^2) \\ Vaclav Kotesovec, Sep 23 2023

A355922 Decimal expansion of Sum_{k>=2} (1/k)*arctanh(1/k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 21 2022

			0.67656513614796664082647201...

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 564.

Cf. A016655, A355921, A355923.

RealDigits[N[Sum[ArcTanh[1/k]/k, {k, 2, Infinity}], 30], 10, 26][[1]] (* Amiram Eldar, Jul 21 2022 *)

Equals Sum_{k>=2} arccoth(k)/k.
Equals Sum_{k>=1} (zeta(2*k)-1)/(2*k-1).
Equals log(Product_{k>=2} ((k+1)/(k-1))^(1/(2*k))).

More terms from Jinyuan Wang, Jul 21 2022

cp's OEIS Frontend

A231132 Decimal expansion of sum_(n=2..infinity) (-1)^n*zeta(n)/n^2.

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