i equally thanked everybody once! PaulRS' solution is nice and my solution differs from his in the last setp only, where i applied Cesaro's mean, as Moo mentioned, to the sequence
that will give you and thus
Define , then
If exists, it implies:
(2) Series , converges, since .
The first term converges since it is . The second term is .
For the second term to converge, it requires , so , noting . This also confirms does exist.