Limit of a Sequence 03

Calculus Level 2

Let $a_n$ be a real-valued sequence such that $\displaystyle{\lim_{n \rightarrow \infty} a_n = 0}$ . Which of the following is the STRONGEST true statement that can be made about $\displaystyle{\frac{a_{n+1}}{a_n}}$ ?

None of these are necessarily true.

\displaystyle \lim_{n \to \infty} \frac{a_{n+1}}{a_n}

exists and is equal to 0.

\displaystyle \lim_{n \to \infty} \frac{a_{n+1}}{a_n}

exists and is infinite.

\displaystyle \lim_{n \to \infty} \frac{a_{n+1}}{a_n}

does not exist or is undefined.

1 solution

Daniel Juncos
Sep 17, 2017

Let $\displaystyle{a_n = \frac{1}{n}}$ . Then $\displaystyle{\lim_{n \rightarrow \infty} a_n = 0}$ , and $\displaystyle{\frac{a_{n+1}}{a_n} = \frac{\frac{1}{n+1}}{\frac{1}{n}} = \frac{n}{n+1} \rightarrow 1}$ as $n \rightarrow \infty$ .

Limit of a Sequence 03

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