Conditions for $\frac{a_{n+1}}{a_n}$ to converge? Given a sequence $a_n \to 0$. I'm looking for a necessary and sufficient condition for $\frac{a_{n+1}}{a_n}$ to converge.
In general, the form $\lim _{n\rightarrow \infty}\frac{a_{n+1}}{a_n}$ will converge if the limit of this is $&lt;1$. If the above holds, then write $a_{n+1}&lt;a_n$ for every index $n$. This is the definition of a monotonically decreasing function. 
A necessary condition would be that $a_n \neq 0.$ Monotonically decreasing would be a sufficient condition. Also, non-negativity of the sequence follows from $a_n\neq 0$ since $a_n=0\Rightarrow  a_{n+1}&lt;0.$
In general, the form $\lim _{n\rightarrow \infty}\frac{a_{n+1}}{a_n}$ will converge if the limit of this is $&lt;1$. If the above holds, then write $a_{n+1}&lt;a_n$ for every index $n$. This is the definition of a monotonically decreasing function. 
A necessary condition would be that $a_n \neq 0.$ Monotonically decreasing would be a sufficient condition. Also, non-negativity of the sequence follows from $a_n\neq 0$ since $a_n=0\Rightarrow  a_{n+1}&lt;0.$