Use the Alternating Series Test to determine if the following series is conditionally convergent, absolutely convergent, or divergent.
${\sum }_{n=1}^{\infty }{(-1)}^{n-1}\left(\frac{3}{4n+5}\right)$

• A. Conditionally convergent
• B. Absolutely convergent
• C. Divergent

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the Alternating Series Test (Leibniz Criterion). It states that an alternating series ∑(-1)^(n-1)a_n converges if: (1) The terms a_n are positive, (2) The terms a_n are decreasing, and (3) The limit of a_n as n approaches infinity is 0.
2. Step 2: Identify the sequence a_n in the given series. Here, a_n = 3 / (4n + 5). Verify that a_n is positive for all n ≥ 1, which is true since the numerator and denominator are positive.
3. Step 3: Check if a_n is decreasing. To do this, compare a_n and a_(n+1): a_n = 3 / (4n + 5) and a_(n+1) = 3 / (4(n+1) + 5). Since the denominator increases as n increases, a_n decreases as n increases.
4. Step 4: Evaluate the limit of a_n as n approaches infinity. Compute lim(n→∞) 3 / (4n + 5). Since the denominator grows without bound, the limit is 0, satisfying the third condition of the Alternating Series Test.
5. Step 5: Since all three conditions of the Alternating Series Test are satisfied, the series ∑(-1)^(n-1)(3 / (4n + 5)) is convergent. However, to determine if it is absolutely convergent, check the series ∑|a_n| = ∑(3 / (4n + 5)). This series does not converge because it resembles a harmonic series, which diverges. Therefore, the series is conditionally convergent.