A263192 -id:A263192 - cp's OEIS frontend

A113024 Decimal expansion of Sum_{k>=1} -(-1)^k/sqrt(k).

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

Offset: 0

Robert G. Wilson v, Oct 11 2005

			1 - 1/sqrt(2) + 1/sqrt(3) - 1/sqrt(4) + 1/sqrt(5) - 1/sqrt(6) + 1/sqrt(7) ... =
0.60489864342163037024726591423595549975976254513024738037854664808...

Stephen Fletcher Hewson, A Mathematical Bridge: An Intuitive Journey In Higher Mathematics, World Scientific, NJ, 2003, p. 83.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Rick Kreminski, Using Simpson's rule to approximate sums of infinite series, Coll. Math. J. 28 (5) (1997), p 368-376, Table 1.
Eric Weisstein's World of Mathematics, Zeta Function.

Cf. A002193, A059750, A263192, A263193, A265162.

Zeta(0,1/2,1/2); evalf(%) ; # R. J. Mathar, Dec 17 2024

RealDigits[(1 - Sqrt[2])Zeta[1/2], 10, 111][[1]]

(1-sqrt(2))*zeta(1/2) \\ G. C. Greubel, Apr 09 2018

Equals (1-sqrt(2))*zeta(1/2) = (-1+A002193) * A059750.
A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 03 2015
Equals -zeta(1/2, 1/2). - Peter Luschny, Nov 03 2020

A263193 Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Oct 11 2015

A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).

			1.043982102849161527453294872586750459790907144722612...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT].

Cf. A096444, A113024, A263192.

evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) - Zeta(0, 1/2, 1-1/(2*Pi))), 120);

N[(Zeta[1/2, 1/(2*Pi)] - Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
RealDigits[Re[(1/2)*I*(PolyLog[1/2, E^(-I)] - PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)

(Zeta(1/2, 1/(2*Pi)) - Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.

A122143 Decimal expansion of Sum_{k >= 1} cos(k)/k^2.

Original entry on oeis.org

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

Offset: 0

T. D. Noe, Aug 28 2006

Also, decimal expansion of the real part of Sum_{k>=1} e^(i*k)/k^2. [Bruno Berselli, Mar 24 2013]

			0.324137740053329817241093475006273747120365201519245527248085933216...

Cf. A096418 (decimal expansion of Sum_{k >= 1} sin(k)/k^2).

Cf. A263192, A121225.

Print[x=FullSimplify[Sum[Cos[n]/n^2, {n,Infinity}]]]; RealDigits[N[x,110]][[1]]

(2*Pi*(Pi-3)+3)/12 \\ Jianing Song, Nov 09 2019

Equals (2*Pi*(Pi-3)+3)/12.

A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 09 2019

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

			0.65847894846240835431252317365398422201349098573375...

Cornel Ioan Vălean, Problem 11930, The American Mathematical Monthly, Vol. 123, No. 8 (2016), p. 831; A Telescoping Series with Inverse Hyperbolic Sine, Solution to Problem 11930 by Ángel Plaza, ibid., Vol. 125, No. 6 (2018), pp. 568-569.

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
A329246 (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

Digits := 100: (log(2 + sqrt(3))/2)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 09 2019

RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* Amiram Eldar, Dec 05 2021 *)

default(realprecision, 100); log(2 + sqrt(3))/2

Equals log(2 + sqrt(3))/2.
Equals -log(2*sin(Pi/12)).
Equals arccoth(sqrt(3)). - Amiram Eldar, Dec 05 2021
From Amiram Eldar, Mar 26 2022: (Start)
Equals arcsinh(1/sqrt(2)).
Equals Sum_{n>=1} arcsinh(1/(sqrt(2^(n+2)+2)+sqrt(2^(n+1)+2))) (Vălean, 2106). (End)
log(2 + sqrt(3))/2 = Sum_{n >= 1} 1/(n*P(n, sqrt(3))*P(n-1, sqrt(3))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log(2 + sqrt(3))/2  = 0.658478948(35...) correct to 9 decimal places. - Peter Bala, Mar 16 2024

A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 09 2019

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

			Sum_{k>=1} cos(k*Pi/4)/k = -log(2*|sin(Pi/8)|) = 0.2673999983...

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
A329247 (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

default(realprecision, 100); log(1 + sqrt(2)/2)/2

Equals log(1 + sqrt(2)/2)/2.

cp's OEIS Frontend

A113024 Decimal expansion of Sum_{k>=1} -(-1)^k/sqrt(k).

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A263193 Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).

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A122143 Decimal expansion of Sum_{k >= 1} cos(k)/k^2.

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A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

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A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

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