Thread: Quick question on series

If I know that a series
$\displaystyle
\sum a_n
$
is convergent. Is it not possible to further deduce that
$\displaystyle
\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} \leq 1
$

Hjörtur

Originally Posted by hjortur

If I know that a series
$\displaystyle
\sum a_n
$
is convergent. Is it not possible to further deduce that
$\displaystyle
\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} \leq 1
$

Hjörtur

The implication in the other direction is straightforward: it's the Ratio Test:

Ratio Test - ProofWiki

Does that help at all?

I just wanted to be sure, I am pretty sure that the limit must hold.
What if $\displaystyle a_n=0$ for all n. What happens to the limit then? Undefined?

Originally Posted by hjortur

I just wanted to be sure, I am pretty sure that the limit must hold.
What if $\displaystyle a_n=0$ for all n. What happens to the limit then? Undefined?

Looks like it to me. The ratio test definitely holds in the other direction (as I've said) but you've found a counterexample such that the way you stated it does *not* hold.

Then i'll just assume that particular series is not 0. Thanks