$a^{n^2}x^n$ which convergence test?

Question

I've been trying to test $a^{n^2}x^n$ for convergence with the ratio, root, leibniz and direct comparison tests but don't seem to get loose of the $n$. any hints?

Answer 1

Use Root test. In fact,

$$ b_n = a^{n^2} = a^{nn} = (a^{n})^n $$

$$ \therefore \liminf | b_n |^{1/n} = \liminf a^n $$

According to the values of $a$, this limit will vary. In fact, If $|a| < 1$, then the limit is $0$ which implies that your series converges when $|a| < 1 $. If $a > 1 $, then the limit is obviously $\infty$ as you should check. Hence, it diverges if $a > 1 $.

Answer 2

I think you can use the root test here.

$$r=\lim\sup_{n\rightarrow \infty}\sqrt[n]{|a^{n^2}x^n|}=\lim\sup_{n\rightarrow \infty}{|a|^n |x|}=|x|\lim\sup_{n\rightarrow \infty}{|a|^n}$$

Therefore by the root test the series is convergent, divergent and inconclusive (may converge or diverge) when, $x<\frac{1}{\lim\sup_{n\rightarrow \infty}{|a|^n}},\, x>\frac{1}{\lim\sup_{n\rightarrow \infty}{|a|^n}}$ and $x=\frac{1}{\lim\sup_{n\rightarrow \infty}{|a|^n}}$ respectively.