Solutions
  1. ak = 1  (the partial sums starting with k=1).
     
  2. ak = (-1)k  (the partial sums starting with k=0).
     
  3. 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.
     
  4. The given series is defined by ak = 1/3k  (the partial sums starting with k=0). The exact limit is 3/2.
     
  5. It converges pretty fast because kk rapidly grows with increasing k, the items 1/kk of the sums thus increasing very fast.
     
  6. The limit of the series is the number p.