Can we say that $\lim\limits_{n \to \infty} a_{n} b_{n} = \infty $ if one limit (either $\lim\limits_{n \to \infty} a_{n} $ or $\lim\limits_{n \to \infty} b_{n} $) doesn't exist ($\pm \infty$) and the other limit is non-zero? Because $\lim\limits_{n \to \infty} a_{n} b_{n} = \lim\limits_{n \to \infty} a_{n} \cdot \lim\limits_{n \to \infty} b_{n}$ is true only when both limits exist.
The context for this confusion:
Q) What is the domain of the power series $\sum_{n=0}^\infty n^nx^n$. (For what values of $x$ does it converge)
I saw a couple solutions online that used the Ratio test in the following way:
$$\text{If}\lim\limits_{n \to \infty} \left \lvert {a_{n+1} \over a_{n}} \right \rvert < 1 \text{ then } \sum a_{n} \text{ converges.}$$
So,
$$\lim\limits_{n \to \infty} \lvert {a_{n+1} \over a_{n}} \rvert = \lim\limits_{n \to \infty} \left \lvert{(n+1)^{n + 1} x^{n+1} \over n^{n}x^{n}} \right \rvert$$
$$ = x\lim\limits_{n \to \infty} \left \lvert{\left ({1 + {1 \over n}}\right )^{n} (n+1)}\right \rvert$$
Now, $\lim\limits_{n \to \infty} \left(1 +{1 \over n} \right)^n = e$ and $\lim\limits_{n \to \infty} n+1 = \infty$
The limit of the first term exists (finite non-zero number), but the limit of the second term doesn't.
So, can we conclude that $\lim\limits_{n \to \infty} \left \lvert {a_{n+1} \over a_{n}} \right \rvert = \infty$?