OFFSET
0,1
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... .
EXAMPLE
0.60211234931037155497112632...
MATHEMATICA
y = NIntegrate[(1 - 2^(1-x)) Zeta[x], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]
PROG
(PARI) intnum(x=0, 1, (1-2^(1-x))*zeta(x)) \\ Michel Marcus, Nov 14 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Alejandro Malla, Nov 13 2023
STATUS
approved