OFFSET
0,1
COMMENTS
The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (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 and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .
EXAMPLE
Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .
MATHEMATICA
f[x_] := (1 - 2^(1 - x)) Zeta[x];
c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling and Peter J. C. Moses, Nov 13 2023
STATUS
approved