A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

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

Offset: 0

Author

Jianing Song, Nov 09 2019

Keywords

• nonn
• cons

Comments

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.

In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:

(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;

(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;

(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.

As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

Examples

			Sum_{k>=1} cos(k*Pi/4)/k = -log(2*|sin(Pi/8)|) = 0.2673999983...
		

Crossrefs

Similar sequences:

A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));

A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));

A329247 (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));

A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));

this sequence (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));

A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));

A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));

A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

Programs

Mathematica

RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

PARI

default(realprecision, 100); log(1 + sqrt(2)/2)/2

Formula

Equals log(1 + sqrt(2)/2)/2.