Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 2 / b

False. Take the positive sequence $\boxed{a_n=n}$. Thenbut $a_n\to +\infty$, so it has no finite limit of a sequence. Successive ratios close to one do not prevent cumulative growth.

False. Take the positive geometric sequence $\boxed{a_n=2^{-n}}$. It tends to zero, whileA positive limit of a sequence would make the claim true since $a_{n+1}/a_n$ would then tend to $\ell /\ell =1$, but positivity of each individual term does not exclude a zero limit of a sequence.

True. Since $a_n^n\to 2$, there is $N_0$ such that $1<a_n^n<3$ for $n\ge N_0$. Positivity then gives $a_n>1$ on this tail.

Fix $\epsilon >0$. If $a_n\ge 1+\epsilon$, Bernoulli's inequality givesFor $n>2/\epsilon$, this contradicts $a_n^n<3$. Therefore eventually $1<a_n<1+\epsilon$, proving

True. Choose $N$ such that $\ell /2<a_n<3\ell /2$ for every $n\ge N$. All the finitely many preceding terms are also positive. Define the positive minimum and finite maximumEvery term lies between $\eta$ and $B$, so every ratio is at most $B/\eta$. Consequently the required supremum satisfies

False. Choose the positive sequenceIt has limit of a sequence one, but Bernoulli's inequality givesThus $(a_n^n)$ is not a bounded sequence.