# Find $\lim\limits_{n\to\infty}\left(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+…+\frac{a_n}{a_1}\right)$

Find $\lim\limits_{n\to\infty}\left(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+...+\frac{a_n}{a_1}\right)$ if {$a_n$} is random sequence with positive terms.

If sequence is increasing ($a_1>a_2>...>a_n$), then $L=+\infty$ What is the limit when sequence is decreasing?

Using AM $\ge$ GM inequality,we get that $\left(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+...+\frac{a_n}{a_1}\right) \ge n$.Can you conclude now?