Pointwise convergence domain of function series

Question

I'm stuck at finding pointwise convergence domain of the following function series $$\sum_{n=1}^\infty \frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}$$

I tried to use d'Alembert and Weierstrass tests, but it seems to me they don't work here.

Answer 1

We have $$(n+1)^{1/3}-n^{1/3}=n^{1/3}\left(\left(1+\frac{1}{n}\right)^{1/3}-1\right)\sim_\infty\frac{1}{3}n^{-2/3}$$ so there's two cases:

• If $x>0$ then $$\frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}\sim_\infty \frac{1}{3}\frac{1}{n^{x+2/3}}$$ and hence the given series is convergent if $x+2/3>1\iff x>\frac{1}{3}$
• If $x\leq 0$ then $$\frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}\sim_\infty \frac{C}{n^{2/3}}$$ where $C=1/2$ if $x=0$ and $C=1$ if $x<0$ and the series is divergent so we conclude that we have the convergence of the series if and only if $x>1/3$.