 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-ii/2/b/i/solution

False. Take the positive sequence

a_{n} = n

. Then

\frac{a _{n + 1}}{a _{n}} = \frac{n + 1}{n} = 1 + \frac{1}{n} ⟶ 1,

but

a_{n} \to + \infty

, so it has no finite limit of a sequence. Successive ratios close to one do not prevent cumulative growth.

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