A269574 -id:A269574 - cp's OEIS frontend

A093720 Decimal expansion of Sum_{n >= 2} zeta(n)/n!.

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

Offset: 1

Eric W. Weisstein, Apr 12 2004

			1.078188729575818482758265436769832381707219...

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A076813, A093721, A269574, A269611, A269720, A269768.

evalf(Sum(exp(1/n)-1-1/n, n=1..infinity), 120); # Vaclav Kotesovec, Mar 04 2016

digits = 99; ClearAll[z, rd]; z[k_] := z[k] = z[k-1] + N[Sum[Zeta[n]/n!, {n, 2^(k-1) + 1, 2^k}], digits]; z[0] = 0; rd[k_] := rd[k] = RealDigits[z[k]][[1]]; rd[0]; rd[k = 1]; While[ rd[k] != rd[k-1], k++]; rd[k] (* Jean-François Alcover, Nov 09 2012 *)

suminf(n=2, zeta(n)/n!) \\ Michel Marcus, Mar 15 2017

Equals Sum_{k>=1} (exp(1/k) - 1 - 1/k). - Vaclav Kotesovec, Mar 04 2016

Equals Integral_{x=0..oo} exp(1/(x^2 + 1))*sin(x/(x^2 + 1))*(coth(Pi*x) - 1) dx + A091725 - 2*A001620 - exp(1)/2 + 3/2. - Velin Yanev, Nov 14 2024

Corrected by Fredrik Johansson, Mar 19 2006

A269611 Decimal expansion of Sum_{n>=1} (sin(Pi/n))^2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 01 2016

			4.32267504323963714111855606344042809207852173550531955525699965992300301...

Cf. A051762, A085365, A093721, A269574, A269720.

evalf(Sum((sin(Pi/n))^2, n=1..infinity), 120);

RealDigits[NSum[Sin[Pi/n]^2, {n, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]

default(realprecision,120); sumpos(n=1, (sin(Pi/n))^2)

Equals (1/2) * Sum_{n>=1} (1 - cos(2*Pi/n)).
Equals Sum_{k>=1} (-1)^(k+1) * 2^(2*k-1) * Pi^(2*k) * Zeta(2*k) / (2*k)!, where Zeta is the Riemann zeta function.
Equals Sum_{k>=1} 2^(4*k-2) * Pi^(4*k) * B(2*k) / (2*k)!^2, where B(n) is the Bernoulli number A027641(n)/A027642(n).

A093721 Decimal expansion of Sum_{n>=1} zeta(2n)/(2n)!.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 12 2004

			0.86900199196290899881105480556139568889249484188057785071064577856...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A076813, A093720, A269574, A269611.

evalf(Sum(cosh(1/n)-1, n=1..infinity), 120); # Vaclav Kotesovec, Mar 04 2016

digits = 105; z[k_] := z[k] = z[k-1] + N[Sum[Zeta[2n]/(2n)!, {n, 2^(k-1) + 1, 2^k}], digits]; z[0] = N[Pi^2/12, digits]; rd[k_] := rd[k] = RealDigits[z[k]][[1]]; rd[0]; rd[k = 1]; While[rd[k] != rd[k-1], k++]; rd[k] (* Jean-François Alcover, Nov 09 2012 *)

suminf(n=1, zeta(2*n)/(2*n)!) \\ Michel Marcus, Mar 20 2017

Equals Sum_{k>=1} (cosh(1/k) - 1). - Vaclav Kotesovec, Mar 04 2016

A269720 Decimal expansion of Sum_{n>=1} (Pi/n - sin(Pi/n)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 04 2016

			4.096434891501739832220234588626055495928144165119120475644486640639751...

Cf. A269574, A269611, A093720, A269768.

evalf(Sum(Pi/n - sin(Pi/n), n=1..infinity), 120);

RealDigits[NSum[Pi/n - Sin[Pi/n], {n, 1, Infinity}, WorkingPrecision->200, NSumTerms->10000, PrecisionGoal->120, Method->{"NIntegrate", "MaxRecursion"->100}]][[1]]
(* Be aware that N[Sum[Pi/n - Sin[Pi/n], {n, 1, Infinity}], 120] give an incorrect numerical result, only 25 decimal places are correct! *)

default(realprecision,120); sumpos(n=1, Pi/n - sin(Pi/n))

Equals Sum_{k>=2} (-1)^k * Pi^(2*k-1) * Zeta(2*k-1) / (2*k-1)!, where Zeta is the Riemann zeta function.

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A093720 Decimal expansion of Sum_{n >= 2} zeta(n)/n!.

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A269611 Decimal expansion of Sum_{n>=1} (sin(Pi/n))^2.

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A093721 Decimal expansion of Sum_{n>=1} zeta(2n)/(2n)!.

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A269720 Decimal expansion of Sum_{n>=1} (Pi/n - sin(Pi/n)).

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