A093721 -id:A093721

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A093720

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

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

FORMULA

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

EXAMPLE

1.078188729575818482758265436769832381707219...

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Apr 12 2004

EXTENSIONS

Corrected by Fredrik Johansson, Mar 19 2006

STATUS

approved

A269611

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

+10
5

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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).

EXAMPLE

4.32267504323963714111855606344042809207852173550531955525699965992300301...

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 01 2016

STATUS

approved

A269574

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

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Value very close to A193081.

LINKS

Table of n, a(n) for n=1..105.

FORMULA

Equals 2 * Sum_{n>=1} (sin(Pi/(2*n)))^2.

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

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

EXAMPLE

4.87071896189479740325580288922801180768723798317416757630477557161789...

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A051762, A085365, A093721, A193081, A269611, A269720.

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 01 2016

STATUS

approved

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