sing the ratio test, or otherwise, determine whether the series∞Xn=1(−1)n n!nn+1is convergent or divergent. You may use the fact that for all real numbers x,the sequence ((1 + (x/n))n) → ex as n → ∞.

To determine whether the series is convergent or divergent, we can use the Ratio Test. The Ratio Test states that for a series Σa_n, if the limit as n approaches infinity of |a_(n+1)/a_n| is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.

The given series is Σ((-1)^n * n! / n^(n+1)). Let's find the ratio a_(n+1)/a_n:

a_(n+1)/a_n = [(-1)^(n+1) * (n+1)! / (n+1)^(n+2)] / [(-1)^n * n! / n^(n+1)] = [(-1)^(n+1-n) * (n+1)!/n! * n^(n+1) / (n+1)^(n+2)] = -[(n+1) / (n+1)^(2) * n / (n+1)] = -n / (n+1)^2

Now, let's find the limit as n approaches infinity of |a_(n+1)/a_n|:

lim (n→∞) |-n / (n+1)^2| = lim (n→∞) |n / (n^2 + 2n + 1)| = 0

Since the limit is less than 1, by the Ratio Test, the series converges.