A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

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

Offset: 1

Author

Benoit Cloitre, Jan 07 2006

Keywords

• nonn
• cons

Comments

Known to be transcendental. After n=2 it is the same as A100554(n).

Imaginary part of psi(I) (for the real part, see A248177). - Stanislav Sykora, Oct 03 2014

Examples

			2.076674047468581174134050794750000490445656266403816665575062484390...
		

References

• Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008.

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Wikipedia, Digamma function.
• Index entries for transcendental numbers.

Crossrefs

Cf. A013661 (Sum_{i>=1} 1/i^2), A232883 (Sum_{i>=0} 1/(2*i^2+1)). - Bruno Berselli, Dec 02 2013

Cf. A248177.

Essentially the same as A100554 and A259171.

Programs

Mathematica

RealDigits[N[Im[PolyGamma[0,I]],105]][[1]]  (* Vaclav Kotesovec, Oct 03 2014 *)

PARI

1/2+Pi/tanh(Pi)/2

PARI

imag(psi(I)) \\ Stanislav Sykora, Oct 03 2014

PARI

sumnumrat(1/(x^2+1),0) \\ Charles R Greathouse IV, Jan 20 2022

Formula

Equals 1/2 + Pi /(2*tanh(Pi)).

Equals 1+Integral_{x >= 0} sin(x)/(exp(x)-1) dx. - Robert FERREOL, Jan 12 2016.

Equals Sum_{k>=0} (-1)^(k+1)*(zeta(2*k) - 1). - Amiram Eldar, Apr 28 2025

Extensions

Offset changed from 0 to 1 by Bruno Berselli, Dec 02 2013