analysis - The limit of these sequences

In my analysis homework, I have the following two True/False questions about limits:

(c) if

$$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$$

Then $(a_n)$ converges

(d) if

$$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$$

Then $(a_n)$ diverges

The thing is: I'm not sure how to interpret these questions. In the answer key, both are given as False; for (c) they give the example where $a_n = n$, and for (d) they give $a_n=\frac{1}{n}$.

Here's where I get confused:

1. For both (c) and (d), how do I interpret the question exactly? It seems that they give counterexamples where it couldn't be the case that $(a_n)$ converges / diverges.





2. The way I interpret it now is that the limit of the absolute value of $\frac{a_{n+1}}{a_n}$ as n tends to infinity is 1. So if we have $a_n =1$, how is this not the case for the sequence $(a_n)$?

Answer

There's a default convention behind such questions. Informally speaking: any such statement with some object in it is understood to be true if it is always true, i.e. if it is true for any such object.

For example, the statement

(c) If $\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|=1$, then $(a_n)$ converges,

actually is meant to say

(c) For any sequence $(a_n)$, if $\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|=1$, then $(a_n)$ converges,

and that's what you need to classify as true or false.

• The statement is true if it is true for all such sequences $(a_n)$;


• The statement is false if at least one counterexample exists (regardless of the fact that there may be other examples that satisfy it).

The answer key for this question gives you such a counterexample: the sequence $a_n=n$ satisfies the "if" part since $\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty} \frac{n+1}{n}=1$, and yet $\displaystyle \lim_{n\to\infty}a_n$ doesn't exists since these numbers tend to infinity.

Question d) should be interpreted similarly.

And then if you put them together, then as @Peter said, we see that the fact $\displaystyle \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|=1$ gives us no information about the behavior of the sequence $(a_n)$: some such sequences, like the counterexample for c), diverge; and some such sequences, like the counterexample for d), converge.