Question

Proving the Alternating Series Test (Theorem 2.7.7) amounts to showing that the sequence of partial sums snₙ= 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.

Consider the subsequences (s₂ n) and (s₂ n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.

Answer 1

The Alternating Series Test states that if an alternating series satisfies two conditions, then the series converges. By considering subsequences and applying the Monotone Convergence Theorem, a third proof for the Alternating Series Test can be obtained.

Step-by-step explanation:

The Alternating Series Test states that if an alternating series satisfies two conditions:

1. The absolute values of the terms in the series form a decreasing sequence, meaning that |a₁| ≥ |a₂| ≥ |a₃| ≥ ... ≥ 0;
2. The limit of the absolute values of the terms approaches zero, meaning that lim┬(n → ∞)⁡|aₙ| = 0;

Then, the series converges.

By considering the subsequences (s₂n) and (s₂n+1), where sn = a₁ − a₂ + a₃ − · · · ± aₙ, and applying the Monotone Convergence Theorem, a third proof for the Alternating Series Test can be obtained. This proof shows that the subsequences (s₂n) and (s₂n+1) are both bounded and monotonic, which implies that they converge to the same limit. Since both subsequences are convergent, the original sequence sn also converges.