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I am trying to solve this question from a math book. Suppose $a_n$ is convergent, then show whether the following sequence is convergent or divergent: $$ \begin{pmatrix}\cfrac{a_n}{a_{n+1}} \end{pmatrix}_n$$

my initial thoughts were that the series should converge to $1$. But then looking at the d'Alembert's ratio, where it's the reciprocal of this series, I am starting to think that I am wrong with the limit being $1$.

What is ít that I am doing wrong here. Any suggestion would be appreciated.

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    $\begingroup$ Look at $a_n=\frac 1 {n!}$. $\endgroup$ Jun 17, 2021 at 9:48

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If $(a_n)_n$ converges to a nonzero real, then the sequence you've written also converges and converges to $1$. This is due to a more general result: Suppose $(a_n)_n$ and $(b_n)_n$ are convergent sequences such that $\lim_n a_n \neq 0$. Then $$\lim_{n \to \infty} \frac{b_n}{a_n} = \frac{\lim_n b_n}{\lim_n a_n}.$$ ($a_n$ is eventually nonzero and so the limit makes sense.)


However, if $a_n \to 0$, then the situation is more delicate. For example, if $a_n = 1/n$, then the ratio converges to $1$. If $a_n = 1/n!$, then the ratio does not converge (or it diverges to $\infty$).
You can get any limit in between as well.
Indeed, fix any $\alpha \in (1, \infty)$. Define $(a_n)_n$ as $a_n = \alpha^{-n}$. Then, $a_n \to 0$ but $a_n/a_{n + 1} = \alpha$ for all $n$.

You can also get oscillatory behaviour by considering a sequence as $$1, 1, -\frac12, -\frac12, \frac13, \frac13, \ldots.$$

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