 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/3/a/ii/solution

False. Set

a_{n} = 4^{n} / n^{4}

. For

n \geq 4

\frac{a _{n + 1}}{a _{n}} = 4 (\frac{n}{n + 1})^{4} \geq 4 (\frac{4}{5})^{4} = \frac{1024}{625} > 1.

Iteration gives a growing geometric sequence as a lower bound for

a_{n}

. Hence

a_{n} \to \infty

and no eventual constant can bound

4^{n} / n^{4}

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