Question

Determine whether the series is absolutely convergent, conditionally convergent, or divergent: [infinity] (-1)^n(2n)! / (7 · 10 · 13 · ⋯ · (3n + 4)) for n = 1

a) Absolutely convergent
b) Conditionally convergent
c) Divergent
d) Unable to determine

Answer 1

The series is divergent.

Step-by-step explanation:

To determine if the series is absolutely convergent, conditionally convergent, or divergent, we can use the ratio test. Let's find the limit of the absolute value of the ratio of consecutive terms as n approaches infinity.

First, let's simplify the expression inside the absolute value:

|(-1)^n(2n)! / (7 · 10 · 13 · ⋯ · (3n + 4))| = (2n)! / (7 · 10 · 13 · ⋯ · (3n + 4))

Now, let's apply the ratio test:

lim(n → ∞) ((2(n+1))! / (7 · 10 · 13 · ⋯ · (3(n+1) + 4))) / ((2n)! / (7 · 10 · 13 · ⋯ · (3n + 4)))

After simplifying, we get:

lim(n → ∞) (2(n+1))(2(n+1)-1) / (3(n+1)+4) = ∞

Since the limit is infinity, the series is divergent.