Prove the assertion in Theorem 16 that whenever an alternating series satisfying the conditions of Theorem 15 is approximated with one of its partial sums, then the remainder (sum of the unused terms) has the same sign as the first unused term.

THEOREM 15-The Alternating Series Test The series (-1)+u = + Uz s + converges if all three of the following conditions are satisfied: 1. The un's are all positive. 2. The positive u,'s are (eventually) nonincreasing: un Un+1 for all n N, for some integer N. 3. Un 0.