A295219 -id:A295219 - 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

A363502 Decimal expansion of Product_{k>=1} k*sinh(1/k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 30 2023

			1.30797093666428364901210447600705632046551568313822...

Similar constants: A118817, A249673, A295219.

Cf. A027641, A027642.

evalf(exp(sum(log(k*sinh(1/k)), k = 1 .. infinity)), 120)

Block[{$MaxExtraPrecision = 1000}, RealDigits[Exp[Sum[(-1)^(k + 1) * Zeta[2*k]^2 / (k*Pi^(2*k)), {k, 1, 200}]], 10, 120][[1]]]

PARI
```
exp(-sumpos(k=1,-log(k*sinh(1/k))))
```

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

cp's OEIS Frontend

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

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