$\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$ series convergence

Question

$$\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$$

Does this converge or diverge (absolute and/or conditional)?$\\$

I've tried Leibniz, D'Alembert, Cauchy and Cauchy-integral criteria, all that's left is the comparison test (those are the only things I can use). Any hints would be helpful.

Answer 1

For absolute convergence:

Let $a_n=\frac1{(n(n+1))^{1/3}}$. Now,

$$\sum\frac1{(n(n+1))^{1/3}}=\sum\frac1{(n^2+n)^{1/3}}=\sum\frac1{n^{2/3}(1+\frac1n)^{1/3}}$$

Let $b_n=\frac1{n^{2/3}}$ (divergent) $$\lim \frac{a_n}{b_n}=\lim \frac1{(1+\frac1n)^{1/3}}=1\text{ (finite and non-zero)}$$

By Limit Comparison Test, the series is not absolutely convergent.

For conditional convergence:

It is easy to see that $\frac{1}{(n(n+1))^{1/3}}$ is monotonically decreasing and converges to $0$. So by Leibniz's Test it is convergent.

Note: Leibniz test says that $\sum (-1)^na_n$ converges. But you can write this as $\sum(-1)^{n-1}a_{n-1}$. Therefore, it doesn't make a difference that you have $(-1)^{n-1}$ in your series instead of $(-1)^n.$

Answer 2

One may use the alternating series test, just check that $$ n \mapsto \frac{1}{{(n(n+1))^{1/3}}} $$ is monotonically decreasing over $[1,\infty)$ and observe that, as $n \to \infty$, $$ \frac{1}{{(n(n+1))^{1/3}}} \to 0. $$

Answer 3

Leibniz's Test is fine here: the series is alternating and $1/(n(n+1))^{1/3}$ decreases to zero. Hence the series is convergent.