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Decimal expansion of the sum of alternating reciprocal square roots, omitting terms where n is a perfect square.
1

%I #16 Mar 29 2024 21:09:43

%S 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,

%T 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,

%U 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

%N Decimal expansion of the sum of alternating reciprocal square roots, omitting terms where n is a perfect square.

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

%C 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.

%H G. C. Greubel, <a href="/A178678/b178678.txt">Table of n, a(n) for n = 0..10000</a>

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

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

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

%e R=0.0882485371383149391699662072222210683157375892300078737421333614112...

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

%o (PARI) default(realprecision, 100); (sqrt(2)-1)*zeta(1/2)+log(2) \\ _G. C. Greubel_, Jan 27 2019

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

%o (SageMath) numerical_approx((sqrt(2)-1)*zeta(1/2)+log(2), digits=100) # _G. C. Greubel_, Jan 27 2019

%Y Cf. A059750, A002162, A002193.

%K cons,nonn

%O 0,2

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

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