I wonder, Let's say $(a_n)_n$ is a sequence.

If $ \frac{a_{n+1}}{a_n} \to L $ and $ L<1$. Then $ a_n \to 0 $.

But, does it mean that $a_n$ is monotonically decreasing? (For every $n$, $a_n > a_{n+1} $ ) ?

Thank you.

  • 2
    $\begingroup$ Yes, if $a_n>0$ then $\frac{a_{n+1}}{a_n} \to L$ with $L<1$ implies that eventually the sequence is monotonically decreasing. $\endgroup$ – Robert Z Jan 13 at 9:33
  • $\begingroup$ First comment: you seem to consider only positive sequences but you didn't state that $a_n>0$. Secondly, $a_n >a_{n+1}$ for every $n$ is obviously false. You can change the values of $a_1$ and $a_2$ any way you like without changing the value of $\lim \frac {a_{n+1}} {a_n}$. $\endgroup$ – Kavi Rama Murthy Jan 13 at 12:22

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