For which value of $x \in R $ the following series converges
$ \sum_{n=0}^\infty \frac {2x+n}{2+n^3x^2}$
The series of the absolute values is $ \sum_{n=0}^\infty \frac {|2x+n|}{2+n^3x^2}$
and applying the root test: $ lim_{n\rightarrow \infty} \sqrt[n]{ \frac {|2x+n|}{2+n^3x^2}}= lim_{n\rightarrow \infty} \frac {|2x+n|^{ \frac {1} {n}}}{(2+n^3x^2)^{ \frac {1} {n}}}$
$\frac {|2x+n|^{ \frac {1} {n}}}{(2+n^3x^2)^{ \frac {1} {n}}} \sim \frac {1}{(x^2)^{ \frac {1} {n}}} \rightarrow 0 $ if $x \ne 0$
Is it correct?