# Does convergence by the ratio test for sequences implies that the sequence is monotonic?

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.

• 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. – Robert Z Jan 13 at 9:33
• 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 $\lim \frac {a_{n+1}} {a_n}$. – Kavi Rama Murthy Jan 13 at 12:22