Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
A) absolutely convergent
B) conditionally convergent
C) divergent

Question

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. 
A) absolutely convergent 
B) conditionally convergent 
C) divergent

Quizplus · Accepted Answer

To determine whether the series converges or diverges, we use the ratio test.

Taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term, we get:

|((n+1)!)^2 / (2n+3)!| / |(n!)^2 / (2n+1)!| = ((n+1)^2 (2n+1)(2n+2)) / (2n+2)(2n+3) = (n+1)^2 / (2n+3)

As n approaches infinity, the (n+1)^2 term dominates the denominator, and the overall limit approaches zero. Therefore, by the ratio test, the series is absolutely convergent.

Determine Whether the Series Is Absolutely Convergent, Conditionally Convergent, or Divergent