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A178678 Decimal expansion of the sum of alternating reciprocal square roots, omitting terms where n is a perfect square. 1
0, 8, 8, 2, 4, 8, 5, 3, 7, 1, 3, 8, 3, 1, 4, 9, 3, 9, 1, 6, 9, 9, 6, 6, 2, 0, 7, 2, 2, 2, 2, 2, 1, 0, 6, 8, 3, 1, 5, 7, 3, 7, 5, 8, 9, 2, 3, 0, 0, 0, 7, 8, 7, 3, 7, 4, 2, 1, 3, 3, 3, 6, 1, 4, 1, 1, 2, 0, 6, 3, 6, 8, 4, 7, 4, 6, 3, 4, 3, 5, 8, 2, 7, 8, 4, 5, 9, 3, 7, 0, 0, 7, 8, 0, 6, 9, 1, 3, 3, 1, 5, 8, 9, 6, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Provides a closed form for the Riemann Zeta Function of one half: Zeta(1/2) = (1 + sqrt(2))(R - log(2)).

The omitted sum of perfect squares equates to the natural logarithm of 2. Giving the alternating sum of all reciprocal square roots as log(2) - R.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

FORMULA

R = Sum_{n>=2} (-1)^n/sqrt(n) for n that are not a perfect square.

R = 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(5) + 1/sqrt(6) - 1/sqrt(7) + 1/sqrt(8) + ...

R = Sum_{n>=2} (-1)^(n+1)*(1-sqrt(n))/n.

EXAMPLE

R=0.0882485371383149391699662072222210683157375892300078737421333614112...

MATHEMATICA

RealDigits[(Sqrt[2] -1)*Zeta[1/2] +Log[2], 10, 100][[1]]

PROG

(PARI) default(realprecision, 100); (sqrt(2)-1)*zeta(1/2)+log(2) \\ G. C. Greubel, Jan 27 2019

(Magma) SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (Sqrt(2)-1)*Evaluate(L, 1/2) +Log(2); // G. C. Greubel, Jan 27 2019

(Sage) numerical_approx((sqrt(2)-1)*zeta(1/2)+log(2), digits=100) # G. C. Greubel, Jan 27 2019

CROSSREFS

Cf. A059750, A002162, A002193.

Sequence in context: A224875 A242588 A105193 * A217459 A185280 A344074

Adjacent sequences: A178675 A178676 A178677 * A178679 A178680 A178681

KEYWORD

cons,nonn

AUTHOR

Matt Rieckman (mjr162006(AT)yahoo.com), Jun 03 2010

EXTENSIONS

Minor correction, simplified description, and additional comments Matt Rieckman (mjr162006(AT)yahoo.com), Jun 28 2010

STATUS

approved

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Last modified January 29 01:41 EST 2023. Contains 359905 sequences. (Running on oeis4.)