Final answer:
The alternating series ∑ₙ=1[infinity](-1)ⁿ+1ln n/n converges.
Step-by-step explanation:
To determine whether the alternating series ∑ₙ=1[infinity](-1)ⁿ+1ln n/n converges or diverges, we can use the Alternating Series Test. This test states that if the terms of an alternating series decrease in absolute value and approach zero as n approaches infinity, then the series converges.
In this series, the absolute value of the terms is ln n/n. To show that the terms decrease in absolute value, we can take the derivative of ln n/n and show that it is negative for n ≥ 3. Since the derivative is negative, the terms of the series are decreasing.
To show that the terms approach zero, we can take the limit as n approaches infinity of ln n/n, which can be found using L'Hospital's Rule. By taking the derivative of the numerator and denominator and applying the rule multiple times, we can see that the limit is zero. Therefore, the terms approach zero.
Since the series satisfies the conditions of the Alternating Series Test, we can conclude that the series converges.