A336405 Decimal expansion of Sum_{n>=1} log(n*sin(1/n)) (negated).

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

Offset

0,1

Comments

As v(n) = log(n*sin(1/n)) ~ -1/(6*n^2) when n -> oo, this series is convergent (zeta(2)/6 ~ 0.2741556778...).

Links

Table of n, a(n) for n=0..86.

Formula

Equals Sum_{n>=1} log(n*sin(1/n)).

Equals log(A295219).

From Amiram Eldar, Jul 30 2023: (Start)

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

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

Example

-0.28055633622915507960203968093919836217450282945971...

Maple

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

Prog

(PARI) sumpos(n=1, log(n*sin(1/n))) \\ Michel Marcus, Jul 20 2020

Crossrefs

Cf. A013661, A233383, A248945, A295219, A336603.

Cf. A027641, A027642.

Sequence in context: A191334 A251794 A321595 * A188924 A011055 A268813

Adjacent sequences: A336402 A336403 A336404 * A336406 A336407 A336408

Keyword

nonn,cons

Author

Bernard Schott, Jul 20 2020

Extensions

More terms from Jinyuan Wang, Jul 21 2020

Status

approved