Proving Convergence or Divergence using Comparison test

Question

I'm asked to prove the convergence or divergence of the below series using the comparison test.

$$\sum_{n=1}^{\infty}\sqrt[3]{(n-1)}-\sqrt[3]{n}$$

I've reduced the above series to $\sqrt[3]{n}(\sqrt[3]{(1-\frac{1}{n}})-1)=U_n$, but after taking $V_n$ as $\sqrt[3]{n}$ and applying the limit, $\lim_{n\to {\infty}}\frac{U_n}{V_n}$ yields me zero, but the value has to be greater than zero for the comparison test to work. So I'm in a dilemma, am I doing something wrong?

Feel free to correct me if I'm wrong in my calculations.

Answer 1

HINT

By binomial expansion prove that

$$\sqrt[3]{n}-\sqrt[3]{(n-1)}\ge \frac13 \frac1{n^{2/3}}$$

Answer 2

HINT

Firstly, remember that $x^{3} - y^{3} = (x-y)(x^{2} + xy +y^{2})$. Hence we get: \begin{align*} \sqrt[3]{n} - \sqrt[3]{n-1} & = \sqrt[3]{n} - \sqrt[3]{n-1}\times\frac{\sqrt[3]{n^{2}} + \sqrt[3]{n(n-1)} + \sqrt[3]{(n-1)^{2}}}{\sqrt[3]{n^{2}} + \sqrt[3]{n(n-1)} + \sqrt[3]{(n-1)^{2}}}\\ & = \frac{1}{\sqrt[3]{n^{2}} + \sqrt[3]{n(n-1)} + \sqrt[3]{(n-1)^{2}}} \geq \frac{1}{3\sqrt{n^{3}}} \end{align*}