11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \), which causes the terms to alternate in sign.
2. Recall the Alternating Series Test (Leibniz Test), which states that an alternating series \( \sum (-1)^k a_k \) converges if two conditions are met: (1) the sequence \( a_k \) is positive, decreasing, and (2) \( \lim_{k \to \infty} a_k = 0 \).
3. Check the sequence \( a_k = \frac{1}{2k + 1} \): it is positive for all \( k \geq 0 \), and as \( k \) increases, \( 2k + 1 \) increases, so \( a_k \) decreases.
4. Evaluate the limit \( \lim_{k \to \infty} \frac{1}{2k + 1} \). Since the denominator grows without bound, the limit is 0.
5. Since both conditions of the Alternating Series Test are satisfied, conclude that the series \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \) converges.

Key Concepts

Alternating Series Test

The Alternating Series Test determines if a series with terms alternating in sign converges. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, though not necessarily absolutely.

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence involves analyzing term behavior and applying tests like the Alternating Series Test or comparison tests to determine if the sum settles to a finite value.

Behavior of the General Term

Examining the general term of a series, such as (−1)^k / (2k + 1), helps assess whether terms decrease in magnitude and approach zero. This behavior is crucial for applying convergence tests and ensuring the series meets necessary criteria for convergence.