An example for a sequence $\{a_n\}$ of positive terms with $\lim_{n \to\infty}a_n^{1/n}=l<\infty $ but $\frac{a_{n+1}}{a_n}$ has no limit

Question

Can we suggest an example of sequence $\{a_n\}$ of positive terms where $\lim_{n \to\infty}a_n^{\frac{1}{n}}$ exists while not the limit $\lim_{n \to\infty}\frac{a_{n+1}}{a_n}$.

Actually, I was looking for the converse of the statement

''If $\{a_n\}$ is a sequence of positive terms and $\lim_{n \to\infty}\frac{a_{n+1}}{a_n}=l<\infty$, then $\lim_{n \to\infty}a_n^{\frac{1}{n}}=l$.''

Another one: https://math.stackexchange.com/q/2202893/42969 – Martin R Oct 28 '22 at 18:24 — Martin R, Oct 28 '22 at 18:24
And this: https://math.stackexchange.com/q/1784869/42969 – Martin R Oct 28 '22 at 18:28 — Martin R, Oct 28 '22 at 18:28

Surb · Accepted Answer · 2022-10-28T18:18:31.223

2

What about $$a_n=\begin{cases}1&n\in 2\mathbb N\\ 2&n\in 2\mathbb N+1\end{cases}\ \ ?$$

The converse statement is true (i.e. for $(a_n)$ being a postive sequence, $\frac{a_{n+1}}{a_n}\underset{n\to \infty }{\longrightarrow } \ell\implies \sqrt[n]{a_n}\underset{n\to \infty }{\longrightarrow } \ell$ is true).

edited Oct 28 '22 at 18:18

answered Oct 28 '22 at 18:15

Surb

55,662

Oh....Nice....thank you.... – Messi Lio Oct 28 '22 at 18:17

An example for a sequence $\{a_n\}$ of positive terms with $\lim_{n \to\infty}a_n^{1/n}=l<\infty $ but $\frac{a_{n+1}}{a_n}$ has no limit

1 Answers1