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 A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k. 1
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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. LINKS Table of n, a(n) for n=0..90. 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. 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 EXAMPLE 0.65847894846240835431252317365398422201349098573375... 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 *) PROG (PARI) default(realprecision, 100); log(2 + sqrt(3))/2 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)))). Sequence in context: A021607 A298172 A225113 * A133618 A194599 A253300 Adjacent sequences: A329244 A329245 A329246 * A329248 A329249 A329250 KEYWORD nonn,cons AUTHOR Jianing Song, Nov 09 2019 STATUS approved

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Last modified August 4 22:06 EDT 2024. Contains 374934 sequences. (Running on oeis4.)