Proving the Alternating Series Test (Theorem 2.7.7) amounts to showing that the sequence of partial sumns Sₙ = 21 - 22 +03 - ... aₙ converges. (The opening example in Section 2.1 includes a typical illustration of (Sₙ).) Different characterizations of completeness lead to different proofs. Prove the Alternating Series Test by showing that (Sₙ) is a Cauchy sequence.