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

Question

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?

Answer 1

The general term of the series is not converging to $0$: as you mentioned it, its absolute value converges to one.

Therefore the series diverges.

Answer 2

An explicit, crude bound on the terms.

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} $.