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

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

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the Alternating Series Test (Leibniz Test). It states that an alternating series ∑(-1)^n a_n converges if: (1) a_n is positive, (2) a_n is decreasing, and (3) lim(n→∞) a_n = 0.
2. Step 2: Identify the sequence a_n in the given series. Here, a_n = (n+1)/ln(n). Check if a_n is positive for n ≥ 2 (since ln(n) is undefined for n=1). For n ≥ 2, ln(n) > 0, so a_n is positive.
3. Step 3: Check if a_n is decreasing. To do this, analyze the derivative of a_n = (n+1)/ln(n) or compare a_n and a_(n+1). If the derivative is negative or a_(n+1) < a_n for all n ≥ 2, then a_n is decreasing.
4. Step 4: Evaluate lim(n→∞) a_n. Compute lim(n→∞) (n+1)/ln(n). If this limit does not equal 0, the series fails the Alternating Series Test and is divergent.
5. Step 5: Since the series fails the Alternating Series Test (lim(n→∞) a_n ≠ 0), conclude that the series is divergent. Absolute convergence and conditional convergence are not applicable.