Solutions
ak =
1
(the partial sums starting with k=1).
ak =
(-1)k
(the partial sums starting with k=0).
The series converges since the sequence of partiual sums increases
monotonically, and ak
is always
(for k£1)
less than a series whose convergence is known:
k/(k4 + 1)
£ k/(k4 + k4) =
1/(2k3).
If one likes, the further estimation
1/(2k3) £
1/(2k2) is possible.
The series definied by both
1/k3 and
1/k2 converge.
The given series is defined by
ak = 1/3k
(the partial sums starting with k=0).
The exact limit is 3/2.
It converges pretty fast because
kk
rapidly grows with increasing k,
the items 1/kk of the sums
thus increasing very fast.
The limit of the series is the number p.