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

sequences-and-series calculus limits

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?


Hint

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?

Note: Monotonicity of given sequence is unimportant here.

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