Proving the Alternating Series Test (Theorem 2.7.7) amountsto showing that the sequence of partial sums                 sn = a1 − a2 + a3 −· · ·±an converges. (The opening example in Section 2.1 includes a typical illustration of (sn).) Different characterizations of completeness lead to different proofs. (a) Prove the Alternating Series Test by showing that (sn) is a Cauchysequence. (b) Supply another proof for this result using the Nested Interval Property(Theorem 1.4.1). (c) Consider the subsequences (s2n) and (s2n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.

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