 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/19/b/i/solution

For the positive terms

a_{n} = n! / 2^{n}

, the ratio is

a_{n + 1} / a_{n} = (n + 1) /2 \to \infty

. The ratio test therefore gives divergence. More directly, the terms increase by a factor at least two for all

n \geq 3

, so they fail the necessary term test for divergence condition

a_{n} \to 0

. Hence

n = 1 \sum \infty \frac{n !}{2 ^{n}} diverges to + \infty.

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