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

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

For

For

Can you pin down better the borderline of divergence ?

Marco Corvi - 2013