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

%I A329246 #17 May 31 2023 04:27:17
%S A329246 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,
%T A329246 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,
%U A329246 0,1,8,6,6,1,8,5,8,9,3,6,5,0,1,5,3,8,7,6,1,4,2
%N A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.
%C A329246 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 A329246 In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
%C A329246 (a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
%C A329246 (b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
%C A329246 (c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
%C A329246 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.
%F A329246 Equals log(1 + sqrt(2)/2)/2.
%e A329246 Sum_{k>=1} cos(k*Pi/4)/k = -log(2*|sin(Pi/8)|) = 0.2673999983...
%t A329246 RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* _Amiram Eldar_, May 31 2023 *)
%o A329246 (PARI) default(realprecision, 100); log(1 + sqrt(2)/2)/2
%Y A329246 Similar sequences:
%Y A329246 A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
%Y A329246 A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
%Y A329246 A329247 (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
%Y A329246 A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
%Y A329246 this sequence (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
%Y A329246 A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
%Y A329246 A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
%Y A329246 A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).
%K A329246 nonn,cons
%O A329246 0,1
%A A329246 _Jianing Song_, Nov 09 2019