A336603 - OEIS

%I #35 Jul 30 2023 02:26:06

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

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

%U 1,7,0,6,3,1,7,1,9,0,3,8,6,8,9,4,2,9,2,1,4,8,1,3,8,6,2,4,0,9,3,3,8,2,0,1,9

%N Decimal expansion of Sum_{n>=1} log(cos(1/n)) (negated).

%C As w(n) = log(cos(1/n)) ~ -1/(2*n^2) when n -> oo, hence the series w(n) is convergent (zeta(2)/2 ~ 0.822467033...).

%D Xavier Merlin, Methodix Analyse, Ellipses, 1997, Exercice 2 p. 119.

%F Equals Sum_{n>=1} log(cos(1/n)) (negated).

%F Equals log(A118817).

%F From _Amiram Eldar_, Jul 30 2023: (Start)

%F Equals Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!), where B(k) is the k-th Bernoulli number.

%F Equals -Sum_{k>=1} (2^(2*k)-1)*zeta(2*k)^2/(k*Pi^(2*k)). (End)

%e -0.945369054726332935266095215408270198116996...

%p evalf(sum(log(cos(1/n)),n=1..infinity),50);

%o (PARI) sumpos(n=1, log(cos(1/n))) \\ _Michel Marcus_, Aug 01 2020

%Y Cf. A013661, A118817.

%Y Cf. A027641, A027642.

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Jul 27 2020