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A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!. 4
1, 0, 0, 8, 3, 3, 3, 6, 0, 8, 9, 0, 7, 2, 9, 0, 2, 8, 9, 9, 7, 6, 4, 5, 3, 6, 6, 7, 3, 5, 4, 8, 3, 8, 7, 8, 6, 0, 7, 1, 0, 7, 7, 2, 8, 1, 5, 7, 9, 5, 4, 3, 1, 0, 2, 0, 0, 3, 0, 5, 9, 0, 7, 4, 9, 2, 7, 0, 7, 5, 5, 0, 4, 8, 4, 8, 1, 1, 1, 0, 8, 4, 1, 1, 4, 8, 5, 5, 9, 4, 1, 6, 1, 7, 0, 0, 6, 5, 7, 8, 1, 9, 2, 5, 2, 6, 8, 9, 9, 1, 9, 4, 6, 9, 7, 5, 7, 7, 4, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

From Vaclav Kotesovec, Feb 24 2016: (Start)

Sum_{k>=0} 1/k! = A001113 = exp(1).

Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).

Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.

Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...

(End)

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. - Bernard Schott, Mar 02 2020

LINKS

Table of n, a(n) for n=1..120.

Eric Weisstein's World of Mathematics, Factorial Sums

FORMULA

Equals Sum_{k>=0} 1/A100734(k).

Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.

Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - Vaclav Kotesovec, Feb 24 2016

Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2020

EXAMPLE

1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...

MAPLE

evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # Vaclav Kotesovec, Feb 24 2016

MATHEMATICA

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]

PROG

(PARI) suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016

CROSSREFS

Cf. A100734.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!).

Sequence in context: A246822 A168356 A021016 * A200230 A069995 A199863

Adjacent sequences:  A269293 A269294 A269295 * A269297 A269298 A269299

KEYWORD

nonn,cons

AUTHOR

Ilya Gutkovskiy, Feb 21 2016

STATUS

approved

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Last modified March 29 05:39 EDT 2020. Contains 333105 sequences. (Running on oeis4.)