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

%I A329247 #34 Dec 14 2024 21:17:01
%S A329247 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,
%T A329247 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,
%U A329247 0,0,9,2,7,0,8,2,2,1,9,8,8,0,3,7,1,0,9,5,0,6,7
%N A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.
%C A329247 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.
%C A329247 In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
%C A329247 (a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
%C A329247 (b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
%C A329247 (c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
%C A329247 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.
%H A329247 Cornel Ioan Vălean, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.8.831">Problem 11930</a>, The American Mathematical Monthly, Vol. 123, No. 8 (2016), p. 831; <a href="https://doi.org/10.1080/00029890.2018.1460990">A Telescoping Series with Inverse Hyperbolic Sine</a>, Solution to Problem 11930 by Ángel Plaza, ibid., Vol. 125, No. 6 (2018), pp. 568-569.
%F A329247 Equals log(2 + sqrt(3))/2.
%F A329247 Equals -log(2*sin(Pi/12)).
%F A329247 Equals arccoth(sqrt(3)). - _Amiram Eldar_, Dec 05 2021
%F A329247 From _Amiram Eldar_, Mar 26 2022: (Start)
%F A329247 Equals arcsinh(1/sqrt(2)).
%F A329247 Equals Sum_{n>=1} arcsinh(1/(sqrt(2^(n+2)+2)+sqrt(2^(n+1)+2))) (Vălean, 2106). (End)
%F A329247 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
%e A329247 0.65847894846240835431252317365398422201349098573375...
%p A329247 Digits := 100: (log(2 + sqrt(3))/2)*10^91:
%p A329247 ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Nov 09 2019
%t A329247 RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* _Amiram Eldar_, Dec 05 2021 *)
%o A329247 (PARI) default(realprecision, 100); log(2 + sqrt(3))/2
%Y A329247 Similar sequences:
%Y A329247 A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
%Y A329247 A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
%Y A329247 this sequence (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
%Y A329247 A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
%Y A329247 A329246 (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
%Y A329247 A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
%Y A329247 A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
%Y A329247 A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).
%K A329247 nonn,cons
%O A329247 0,1
%A A329247 _Jianing Song_, Nov 09 2019