$\text { Determine whether the series } \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { n + 2 } \text { converges conditionally or }$ absolutely, or diverges.
A)The series converges conditionally but does not converge absolutely.
B)The series converges absolutely but does not converge conditionally.
C)The series diverges.
D)The series converges absolutely.

Question

Quizplus · Accepted Answer

We can use the alternating series test to show that the series converges conditionally. The terms of the series approach zero as n approaches infinity, and the series alternates in sign. Additionally, the terms are decreasing in magnitude because the derivative of 1/(n+2) is always negative. However, the series does not converge absolutely because the series obtained by taking the absolute value of the terms is the harmonic series with the first two terms removed, which is known to diverge.