Final answer:
The sequence converges to 0, as determined by the ratio test showing that the limit of a_(n+1) / a_n approaches 0 as n goes to infinity.
Step-by-step explanation:
To determine whether the sequence an = (−3)n / 4n! converges or diverges, we can examine the behavior of its terms as n goes to infinity. Since n! grows much faster than (−3)n, the terms of the sequence will get closer and closer to zero as n increases, implying that the sequence converges to 0.
The limit of the sequence can be evaluated using the ratio test. We look at the limit of the absolute value of an+1 / an as n approaches infinity. If this limit is less than 1, the sequence converges absolutely, and thus, it also converges.
In this case:
- lim n→∞ |an+1 / an| = lim n→∞ |(−3)n+1 / 4n+1!| / |(−3)n / 4n!| = lim n→∞ |−3/4n+1| = 0
Since this limit is 0, which is less than 1, the sequence converges. The limit of the sequence is 0.