Pointwise convergence domain of function series 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. 

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 $x>1/3$.

