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A181049 Decimal expansion of (Pi/2 - log(1+sqrt(2)))/(2*sqrt(2)) = Sum_{k>=0} (-1)^k/(4k+3). 5
2, 4, 3, 7, 4, 7, 7, 4, 7, 1, 9, 9, 6, 8, 0, 5, 2, 4, 1, 7, 9, 9, 7, 5, 0, 8, 3, 6, 3, 2, 3, 0, 2, 7, 1, 1, 0, 0, 1, 4, 8, 0, 0, 5, 4, 9, 9, 8, 6, 7, 7, 6, 5, 1, 4, 3, 6, 3, 1, 7, 0, 6, 2, 8, 2, 1, 4, 6, 9, 3, 4, 6, 8, 6, 3, 9, 2, 7, 1, 4, 8, 5, 8, 8, 0, 8, 1, 3, 3, 0, 2, 2, 7, 7, 8, 2, 3, 4, 0, 6, 3, 5, 6, 3, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..2015

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

FORMULA

Integral_{x=0..1}(x^2 dx)/(1+x^4) = Sum_{k>=0} (-1)^k/(4k+3) = (Pi/2 - log(1+sqrt(2)))/(2*sqrt(2)) = 0.2437474...

From Peter Bala, Sep 23 2016: (Start)

c = 1/2 * Integral_{x = 0..Pi/4} sqrt(tan(x)) dx. Cf. A247719.

Let N be a positive integer divisible by 4. We have the asymptotic expansion 2*((Pi/2 - log(1 + sqrt(2)))/(2*sqrt(2)) - Sum_{k = 0..N/4 - 1} (-1)^k/(4*k + 3)) ~ 1/N - 1/N^2 - 3/N^3 + 11/N^4 + 57/N^5 - - ..., where the sequence of coefficients [1, -1, -3, 11, 57,...] is A212435. This follows from Borwein et al., Lemma 2 with f(x) = 1/x and then set x = N/4 and h = 3/4. An example is given below. Cf. A181048. (End)

EXAMPLE

(Pi/2 - log(1+sqrt(2)))/(2*sqrt(2)) = 0.2437474...

From Peter Bala, Sep 23 2016: (Start)

At N = 100000 the truncated series Sum_{k = 0..N/4 - 1} (-1)^k/(4*k + 3) ) = 0.4874(8)5494(4)9936(4)048(24)99(444)67(625)6... to 32 digits The bracketed numbers show where this decimal expansion differs from that of 2*A181049. The numbers 1, -1, -3, 11, 57, -361 must be added to the bracketed numbers to give the correct decimal expansion to 32 digits: 2*( Pi/2 - log(1+sqrt(2)))/(2*sqrt(2) ) = 0.4874(9)5494(3)9936(1)048(35) 99(501)67(264)6.... (End)

MATHEMATICA

First@ RealDigits[N[(Pi/2 - Log[1 + Sqrt@ 2])/(2 Sqrt@ 2), 105]] (* Michael De Vlieger, Oct 07 2015 *)

PROG

(PARI)

default(realprecision, 106);

eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(4*n+3)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

(PARI) (Pi/2 - log(1+sqrt(2)))/(2*sqrt(2)) \\ G. C. Greubel, Nov 28 2017

(MAGMA) C<i> := ComplexField(); [(Pi(C)/2 - Log(1+Sqrt(2)))/(2*Sqrt(2))]; // G. C. Greubel, Nov 28 2017

CROSSREFS

Cf. A113476, A001586, A093954, A181048, A212435, A247719.

Sequence in context: A054589 A051851 A011171 * A007203 A110412 A266285

Adjacent sequences:  A181046 A181047 A181048 * A181050 A181051 A181052

KEYWORD

cons,nonn

AUTHOR

Jonathan D. B. Hodgson, Oct 01 2010, Oct 05 2010

STATUS

approved

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Last modified July 17 08:42 EDT 2019. Contains 325098 sequences. (Running on oeis4.)