A260230 Decimal expansion of S(Pi), where S(x) is the series Sum_{n>=1} (-1)^(n+1)*coth(n*x)/n.

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

Offset

0,1

Comments

From Vaclav Kotesovec, Jul 21 2015: (Start)

Sum_{n>=1} (-1)^(n+1)*cos(n*x)/n = log(2*(1+cos(x)))/2.

Sum_{n>=1} cos(n*x)/n = -log(2*(1-cos(x)))/2.

(End)

Links

Table of n, a(n) for n=0..103.

A. Dieckmann, Collection of Infinite Products and Series

Jonathan D. Weiss, The Summation of One Class of Infinite Series, Applied Mathematics, 2014, 5, 2816-2822.

Eric Weisstein's MathWorld, Inverse Nome

Formula

S(Pi) = Sum_{n>=1} (-1)^(n+1)*coth(n*Pi)/n = log(2) + 2*Sum_{k>=1} log(1+exp(-2*k*Pi)).

Equals Pi/6 + (1/4)*log(2).

Example

0.69688557073828520043141526091112795605173660015258145035932743442465...

Mathematica

RealDigits[Pi/6 + (1/4)*Log[2], 10, 104] // First

Crossrefs

Sequence in context: A198144 A154394 A126599 * A159691 A118947 A023410

Adjacent sequences: A260227 A260228 A260229 * A260231 A260232 A260233

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 20 2015

Status

approved