A343469 -id:A343469 - cp's OEIS frontend

A343470 Decimal expansion of Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n.

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

Offset: 0

Bernard Schott, Apr 17 2021

The alternating series test shows the series is convergent.

			0.46571230352630363552682770124023013687237216061516248...

Wikipedia, Alternating series test.

Cf. A343469, A265011.

evalf(sum(((-1)^(n-1))*arctan(n)/n, n=1..infinity),120);

sumalt(n=1, (-1)^(n-1)*atan(n)/n) \\ Michel Marcus, Apr 17 2021

Equals Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n.

A355921 Decimal expansion of Sum_{k>=1} (1/k)*arctan(1/k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 21 2022

			1.40586929828778091125539861...

Mathematics Stack Exchange, Converting the sum: Sum_{n=1..oo}(1/n) * cot^(-1)(n) to an integral, 2017.
Mathematics Stack Exchange, Summing an Arctangent Series, 2021.
Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 428.
Eric Weisstein's World of Mathematics, Sine Integral.

Cf. A091007, A265011, A343469, A343470, A352619, A355922.

RealDigits[N[Sum[ArcTan[1/k]/k, {k, 1, Infinity}], 30], 10, 27][[1]]

default(realprecision, 200); sumalt(k=1,(-1)^(k+1)*zeta(2*k)/(2*k-1)) \\ Vaclav Kotesovec, Jul 21 2022

Equals Sum_{k>=1} arccot(k)/k.
Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k)/(2*k-1).
Equals (1/2) * Integral_{x=0..1} (coth(Pi*x)*Pi/x - 1/x^2) dx.
Equals Integral_{x>=0} Si(x)/(exp(x)-1) dx, where Si(x) is the sine integral function.
Equals -Integral_{x>=0} sin(x)*log(1-exp(-x))/x dx.

More terms from Jinyuan Wang, Jul 21 2022

cp's OEIS Frontend

A343470 Decimal expansion of Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n.

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