It's well known that the harmonic series Sum 1/n diverges, while the sum (from n=1 to infinity) Sum 1/n^s with s > 1 converges (and its value is the Riemann zeta function ζ(s).

It is therefore natural to consider the series in which the exponent depends on n,

For k(n)=0 we have the harmonic series, that diverges. For k(n) constant and positive we have a coverging series. The interesting things occur when k(n) is positive, and approaches 0 as n goes to infinity. For example k(n)=1/n, or k(n)=1/ln(n), or k(n)=1/sqrt(ln(n)). In the first two cases the sum is still diverging. In the last one it converges.

Is there a function the rate of decrease of which is the borderline between convergence and divergence ? In other words, is there a function k_o(n) such that if lim_{n -> inf} k(n)/k_o(n) = 0 than the sum diverges, and if the limit is infinity it converges ?

For k(n) = (ln(ln(n)))^-m (m>=0) the sum converges.
For k(n) = (ln(ln(n)))^{- ln(ln(n))} the sum diverges.
For k(n) = (ln(ln(n)))^{- ln(ln(ln(n)))} = exp( - (ln(ln(ln(n))))² ) the sum converges.

Can you pin down better the borderline of divergence ?

Marco Corvi - 2013