An alternating series convergence

Question

I was wondering whether the following series converges or diverges,

$$\sum_{n=1}^\infty (-1)^n \sqrt[n]{a}$$ $$\forall a>0, a\ne1$$

The divergence test cannot be applied, since the sequence does not have a limit.

Answer 1

Note that $\sqrt[n]a \to 1$, hence $(-1)^n \sqrt[n]a \not\to 0$, so the series diverges.

Answer 2

Since $$\lim_{n \to \infty} (-1)^n \sqrt[n]{a} \neq 0$$ the series is not convergent.