According to the quotient rule, if $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \lt 1$, then the sequence $(a_n)$ must be monotonically decreasing (definitively) and converge to $0$. If the limit is greater than $1$, then the sequence must be monotonically increasing and diverge.

I can't understand why the limit would ever be different than $1$, since if $a_n$ converges to $l$, then any subsequence of said sequence is also convergent and converges to the same limit $l$. Consequently, both $(a_n)$ and $(a_{n+1})$ converge to the same limit, and here lies the crux of my question: why can $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \neq 1$ if both sequences converge to the same limit $l$?

Please anyone shed some light on this as I really can't wrap my head around it.

  • $\begingroup$ Both sequences converge to zero. $\endgroup$ – Lord Shark the Unknown Oct 29 '17 at 12:49
  • 1
    $\begingroup$ Have you looked at any examples? Try $a_n=\frac 1{n!}$, say. $\endgroup$ – lulu Oct 29 '17 at 12:50
  • $\begingroup$ I can't see how this would change anything. Isn't the limit of both $l$ anyway? $\endgroup$ – NTAuthority Oct 29 '17 at 12:51
  • 2
    $\begingroup$ $\frac 00$ is not the same as $1$ $\endgroup$ – lulu Oct 29 '17 at 12:51
  • $\begingroup$ @lulu Alright I see what you mean there $\endgroup$ – NTAuthority Oct 29 '17 at 12:57

Of course that $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \neq 1$ generally will happen.

The point is that $a_{n+1}$ and $a_n$ are both functions of $n$ but different ones and $a_{n+1}$ is "step ahead" of $a_n$ so they will take different values at the same $n$´s.

And, sometimes, this speed of convergence is fast enough to ensure $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} \neq 1$.

For example, an example in comments is a nice one, we have $a_n=\frac 1{n!}$, but $a_{n+1}=\frac{1}{(n+1)!}=a_n \cdot \frac {1}{n+1}$.

Sometimes this speed will not be fast enough so we will sometimes have $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 1$

  • $\begingroup$ Thanks a lot! This is just what I needed. $\endgroup$ – NTAuthority Oct 29 '17 at 13:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.