A336603 -id:A336603 - cp's OEIS frontend

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

Bernard Schott, Jul 20 2020

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

			-0.28055633622915507960203968093919836217450282945971...

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

Cf. A027641, A027642.

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

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

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)

More terms from Jinyuan Wang, Jul 21 2020

A342647 Decimal expansion of Sum_{n>=1} log(cos(1/n)) * log(sin(1/n)).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 17 2021

log(cos(1/n)) ~ -1/(2*n^2) when n -> oo, so the series log(cos(1/n)) is convergent (A336603), but
log(sin(1/n)) ~ -log(n) when n -> oo, so the series log(sin(1/n)) is divergent.
However, as log(cos(1/n)) * log(sin(1/n)) ~ log(n)/(2*n^2) when n -> oo, the series log(cos(1/n)) * log(sin(1/n)) is convergent.

			0.588251433968163564741783117942531437228475727762559...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.1.f p. 279.

Cf. A336405, A336603.

sumpos(n=1, log(cos(1/n)) * log(sin(1/n))) \\ Michel Marcus, Mar 18 2021

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

a(4)-a(51) from Jon E. Schoenfield, Mar 18 2021

cp's OEIS Frontend

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

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