Question

Which of the following series is/are convergent by the alternating series test?

I only
I and II only
I and III only
II and III only

Answer 1

I and II only

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.