A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

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

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

			0.65847894846240835431252317365398422201349098573375...
		

Links

• Cornel Ioan Vălean, Problem 11930, The American Mathematical Monthly, Vol. 123, No. 8 (2016), p. 831; A Telescoping Series with Inverse Hyperbolic Sine, Solution to Problem 11930 by Ángel Plaza, ibid., Vol. 125, No. 6 (2018), pp. 568-569.

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))));

this sequence (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))));

A329246 (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

Maple

Digits := 100: (log(2 + sqrt(3))/2)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 09 2019

Mathematica

RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* Amiram Eldar, Dec 05 2021 *)

PARI

default(realprecision, 100); log(2 + sqrt(3))/2

Formula

Equals log(2 + sqrt(3))/2.

Equals -log(2*sin(Pi/12)).

Equals arccoth(sqrt(3)). - Amiram Eldar, Dec 05 2021

From Amiram Eldar, Mar 26 2022: (Start)

Equals arcsinh(1/sqrt(2)).

Equals Sum_{n>=1} arcsinh(1/(sqrt(2^(n+2)+2)+sqrt(2^(n+1)+2))) (Vălean, 2106). (End)

log(2 + sqrt(3))/2 = Sum_{n >= 1} 1/(n*P(n, sqrt(3))*P(n-1, sqrt(3))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log(2 + sqrt(3))/2 = 0.658478948(35...) correct to 9 decimal places. - Peter Bala, Mar 16 2024