Proving the Alternating Series Test (Theorem 2.7.7) amounts toshowing that the sequence of partial sums sn = a1 ? a2 + a3 ?· ··±an converges. (The opening example in Section 2.1 includes atypical illustration of (sn).) Different characterizations ofcompleteness lead to different proofs.

(a) Prove the Alternating Series Test by showing that (sn) is aCauchy sequence.

(c) Consider thesubsequences (s2n) and (s2n+1), and show how the MonotoneConvergence Theorem leads to a third proof for the AlternatingSeries Test.