A367309 Decimal expansion of area under the curve (1-2^(1-x))*zeta(x) from 0 to 1.

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

Offset: 0

Author

Alejandro Malla, Nov 13 2023

Keywords

• nonn
• cons

Comments

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to g(x) = (1 - 2^(1-x))*zeta(x) on the open interval (0, 1). This series can be described as an alternating version of the 'p-series' when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x. Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined, but has the limit value log(2). Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

Examples

			0.60211234931037155497112632...
		

Crossrefs

Cf. A113024, A367310, A367311.

Programs

Mathematica

y = NIntegrate[(1 - 2^(1-x)) Zeta[x], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]

PARI

intnum(x=0, 1, (1-2^(1-x))*zeta(x)) \\ Michel Marcus, Nov 14 2023