OFFSET
0,5
LINKS
Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Carleman's Inequality
Wikipedia, Carleman's inequality
FORMULA
sum_{k >= 1} (a_1 a_2 ... a_k)^(1/k) < e*sum_{k >= 1} (1-sum_{j=1..m} b_j/(k+1)^j)*a_k, where a_k >= 0 for all k and a_l > 0 for at least one l, m being any positive integer.
b(0) = -1, b(n) = (-1/n)*sum_{k=1..n} b(n-k)/(k+1).
EXAMPLE
Fractions begin -1, 1/2, 1/24, 1/48, 73/5760, 11/1280, 3625/580608, ...
MATHEMATICA
b[0] = -1; b[n_] := b[n] = (-1/n)*Sum[b[n - k]/(k + 1), {k, 1, n}]; Table[b[n] // Numerator, {n, 0, 20}]
CoefficientList[Series[-(1 - x)^(-(1 - x)/x)/E, {x, 0, 20}], x] // Numerator (* Eric W. Weisstein, Apr 13 2018 *)
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Jean-François Alcover, Oct 24 2014
STATUS
approved