A118817 -id:A118817 - cp's OEIS frontend

A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

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

Offset: 1

Vaclav Kotesovec, Nov 03 2014

			2.116465536505484775878572227025831988148089392809082568281348...

Cf. A118817, A230821, A051762, A218504, A246945, A270614.

Cf. A027641, A027642.

evalf(exp(sum(log(cosh(1/n)), n=1..infinity)), 100)

default(realprecision,120); exp(sumpos(k=1, log(cosh(1/k))))

From Amiram Eldar, Jul 30 2023: (Start)
Equals exp(Sum_{k>=1} 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 exp(Sum_{k>=1} (-1)^(k+1)*(2^(2*k)-1)*zeta(2*k)^2/(k*Pi^(2*k))). (End)

A295219 Decimal expansion of Product_{n>=1} n*sin(1/n).

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Nov 17 2017

			0.75536338851857321406336498617047655...

Cf. A118817.

Cf. A027641, A027642.

evalf(Product(n*sin(1/n), n = 1..infinity), 120); # Vaclav Kotesovec, Jun 23 2021

\\ Set the precision at least twice than the
  \\ number of desired correct decimal digits
  default(realprecision, 200);  \\ To get the first 100 digits right
  exp(-sumpos(n=1, -log(n*sin(1/n))))

Equals 1*sin(1/1) * 2*sin(1/2) * 3*sin(1/3) * 4*sin(1/4) * 5*sin(1/5) * ...
From Amiram Eldar, Jul 30 2023: (Start)
Equals exp(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 exp(-Sum_{k>=1} zeta(2*k)^2/(k*Pi^(2*k))). (End)

Terms corrected by Jinyuan Wang, Jul 21 2020

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

Original entry on oeis.org

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

Bernard Schott, Jul 27 2020

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...).

			-0.945369054726332935266095215408270198116996...

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

Cf. A013661, A118817.

Cf. A027641, A027642.

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

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

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)

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

A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

A295219 Decimal expansion of Product_{n>=1} n*sin(1/n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula