a) n2/2n≥0 for all n, and n2/2n=0 only when n=0. When n=1 n2/2n=½. n=0 establishes the infimum (lower bound) for the series. The monotonic convergence theorem states that if a series has progressively decreasing terms, and is bounded below it will converge to the lower bound. So the series converges to zero.
b) When n=0 10n/(2n+1)!=1/1=1. When n=1 it is 10/6=5/3; when n=2 it's 100/120=⅚. The series is decreasing so the lower bound is 1 (at n=0) and the series convergences to 1 according to the theorem.