A336603 Decimal expansion of Sum_{n>=1} log(cos(1/n)) (negated).

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, 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, 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

Offset: 0

Author

Bernard Schott, Jul 27 2020

Keywords

• nonn
• cons

Comments

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...).

Examples

			-0.945369054726332935266095215408270198116996...
		

References

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

Crossrefs

Cf. A013661, A118817.

Cf. A027641, A027642.

Programs

Maple

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

PARI

sumpos(n=1, log(cos(1/n))) \\ Michel Marcus, Aug 01 2020

Formula

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

Equals log(A118817).

From Amiram Eldar, Jul 30 2023: (Start)

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.

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