Answer:
I and II only
Step-by-step explanation:
According to the alternating series test:
For a series ∑(-1)ⁿ aₙ or ∑(-1)ⁿ⁺¹ aₙ
If lim(n→∞) aₙ = 0
and aₙ is decreasing,
then the series converges.
(I) aₙ = 1 / ln(n)
lim(n→∞) 1 / ln(n) = 0.
1 / ln(n) > 1 / ln(n+1).
This series converges.
(II) aₙ = n / (n²+1)
lim(n→∞) n / (n²+1) = 0.
n / (n²+1) > (n+1) / ((n+1)²+1) for n ≥ 1.
This series converges.
(III) aₙ = 2n / (5n−1)
lim(n→∞) 2n / (5n−1) = 2/5.
The alternating series test is inconclusive for this series.
