The convergence of a series does not depend on any nite number of the terms, though of course the value of the limit does. More precisely, consider series 2 an and Z b,n and suppose the set {71 E N : an 7E 5\"} is nite. Then the series both converge or else they both diverge. Prove this. Hint: This is almost obvious from Theorem 14.4. 14.4 Theorem. A series converges if and only if it satisfies the Cauchy criterion