Question

If L = lim |sn+1/sn| as n goes to infinity, what does L < 1 imply?

A. The series ∑ sn converges absolutely.
B. The series ∑ sn converges conditionally.
C. The series ∑ sn diverges.
D. The series ∑ sn alternates.

Answer 1

If L = lim |sn+1/sn| as n goes to infinity, L < 1 implies that the series ∑ sn converges absolutely.

Step-by-step explanation:

If L = lim |sn+1/sn| as n goes to infinity, L < 1 implies that the series ∑ sn converges absolutely (option A). To understand why, let's break it down step by step:

1. Consider the limit L = lim |sn+1/sn| as n goes to infinity.
2. If L < 1, it means that the absolute value of the ratio sn+1/sn approaches a value less than 1 as n becomes larger and larger.
3. When the ratio of consecutive terms approaches a value less than 1, it implies that the terms of the series sn are decreasing in magnitude.
4. For a series to converge absolutely, the terms must decrease in magnitude. In this case, since the terms of sn are decreasing, the series ∑ sn converges absolutely.

Therefore, the correct answer is option A: The series ∑ sn converges absolutely when L < 1.