# Convergent or Divergent Series? $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt[n+1]{10}}$

I need to find whether this series is convergent or divergent: $$\sum_{n = 1}^{\infty}\frac{\left(-1\right)^{ n + 1}} {\,\sqrt[n + 1]{\, 10\, }\, }$$

(1) Alternating series test does not provide any additional information since $$\lim_{n \to \infty} \frac{1}{\sqrt[n+1]{10}} = 1$$ and not $$0$$. A ratio test with its larger series $$\sum\limits_{n=1}^{\infty} \frac{1}{\sqrt[n+1]{10}}$$ results in $$1$$ meaning its inconclusive. Can someone guide me in the right direction?

• Alternating series does not have a no conclusion option, it can readily give you divergence, but the limit is the best way to go as the answer says. Commented Jul 27, 2020 at 16:13

The general term of the series is not converging to $$0$$: as you mentioned it, its absolute value converges to one.
If $$(1+a)^{1/n} = 1+b$$, then, by Bernoulli's inequality, $$1+a =(1+b)^n \ge 1+nb$$ so $$b \le a/n$$ so $$(1+a)^{1/n} \le 1+a/n$$.
Therefore $$\dfrac1{(1+a)^{1/n}} \ge \dfrac1{1+a/n} =\dfrac{n}{n+a} =1-\dfrac{a}{n+a} \to 1$$ as $$n \to \infty$$.
In this case, $$\dfrac1{10^{1/n}} \ge 1-\dfrac{9}{n+9}$$.