Prove divergence of $\sum\limits_{n=1}^\infty\left(\frac1{\sqrt n}-\frac1{\sqrt{n+1}}\right)^{2/3}$

Question

Prove divergence of $\sum\limits_{n=1}^\infty\left(\frac1{\sqrt n}-\frac1{\sqrt{n+1}}\right)^{2/3}$.

I need to prove that this series is divergent, but the ratio test, root test and the divergent majorant $\frac{1}{n}$ are inconclusive, so how would I be able to prove this?

I also tried it being proportional to something, but I couldn't get a divergent majorant out of it, because for large n it always becomes smaller.

Answer 1

Note that $$ \frac 1{\sqrt n} - \frac 1{\sqrt {n+1}} = \frac {\sqrt{n+1} - \sqrt n}{\sqrt {(n+1)n}} = \frac 1 {\sqrt {n(n+1)} \cdot (\sqrt n + \sqrt{n+1})} \sim \frac 1{2n^{3/2}} \quad [n \to \infty] $$ hence the term is asymptotically $(1/2)^{2/3} n^{-1}$, and the series diverges.

Answer 2

$\frac 1 {\sqrt n} -\frac 1 {\sqrt {n+1}} =\frac 1 {\sqrt n \sqrt {n+1}(\sqrt n +\sqrt {n+1})}$. Compare the given series with $\sum \frac 1 n$ which is divergent.

Answer 3

$$\left(\frac1{\sqrt n}-\frac1{\sqrt{n+1}}\right)^{2/3}=\left(\frac{\sqrt{n+1}-\sqrt n}{\sqrt n\sqrt{n+1}}\right)^{2/3}=\left(\frac{1}{\sqrt n\sqrt{n+1}(\sqrt{n+1}+\sqrt n)}\right)^{2/3}>\left(\frac{1}{\sqrt{n+1}\sqrt{n+1}(\sqrt{n+1}+\sqrt{n+1})}\right)^{2/3}=\frac{2^{-2/3}}{n+1}$$

so that the majorant test is conclusive.