<s>[INST] Convergence of the series $\sum \frac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$ 
To prove that nature of the following series : $$\sum \dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}$$
  they use in solution manual :


My questions:


*

*I don't know how to achieve ( * ) could someone complete my attempts for ( * )  and is it correct if i use (**) to prove that the series is convergent : 


$$\fbox{$\dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}-\dfrac{(-1)^{n}}{n}+O\left(\dfrac{1}{n^{\frac{4}{3}}}\right)$}\quad (*)$$
My thoughts :
\begin{align}
\dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}} &=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \left( 1+\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right) \right)^{-1}
\end{align}

note that :
  $$(1+x)^{\alpha}=1+\alpha x+O(x^{2})$$

\begin{align}
\dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}} &=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \left( 1+\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right) \right)^{-1}\\
&=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \left( 1-\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right)+O\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right)^{2} \right)\\
&=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} -\dfrac{(-1)^{n}}{n}+\dfrac{(-1)^{n}}{n^{\frac{4}{3}}} +\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}\times O\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right)^{2} \\
&=\ldots\ldots \\
&= \mbox{ I'm stuc here i  hope someone complete my attempts } 
\end{align} 
Or i should use :

note that :
  $$(1+x)^{\alpha}=1+O(x)$$

\begin{align}
\dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}} &=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \left( 1+\left(\dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \right) \right)^{-1}\\
&=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}} \left( 1+O\left( \dfrac{1}{n^{\frac{1}{3}}}+\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}\right) \right)\\
&=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}+O\left( \dfrac{1}{n}+\dfrac{(-1)^{n}}{n^{\frac{4}{3}}}\right) \\
&=\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}+O\left( \dfrac{(-1)^{n}}{n^{\frac{4}{3}}}\right) \\
\end{align}
$$\fbox{$\dfrac{(-1)^{n}}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}} =\dfrac{(-1)^{n}}{n^{\frac{2}{3}}}+O\left( \dfrac{(-1)^{n}}{n^{\frac{4}{3}}}\right)$}\quad (**) $$
 [/INST] $\dfrac{1}{n^{\frac{2}{3}}+n^{\frac{1}{3}}+(-1)^{n}}-\dfrac{1}{n^{\frac{2}{3}}}+\dfrac{1}{n} = \dfrac{-n(-1)^n+n + n^{2/3}(-1)^n}{n^{5/3}(n^{2/3}+n^{1/3}+(-1)^n}$
The right hand side has $n$ in the numerator and $n^{7/3}$ in the denominator and hence is $O\left(\frac{1}{n^{4/3}}\right)$
</s>