Prove/disprove : $\left(1+\frac{(-1)^{n}}{\sqrt{n}} \right)^{-1}=\left(1-\frac{(-1)^{n}}{\sqrt{n}}+o\left(\frac{1}{n}\right) \right)$ Why we have that 
$$\begin{align*}
\frac{(-1)^n}{\sqrt{n}}\left(1+\frac{(-1)^n}{\sqrt{n}} \right)^{-1} & =\frac{(-1)^n}{\sqrt{n}}\left(1-\frac{(-1)^n}{\sqrt{n}}+O\left(\frac{1}n\right) \right) \tag{$E_1$} \\
\end{align*}$$
insted of $$\left(1-\frac{(-1)^n}{\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right) \right) \tag{$E_2$} ?$$
Is that because we have $\frac{1}{\sqrt{n}}=o\left(\frac{1}{n}\right)$? But this is wrong! 


*

*Could someone explain to me why we have $E_1$ instead of $E_2$




By Taylor expansion,
\begin{align*}
\frac{1}{1+\frac{(-1)^n}{\sqrt{n}}} &= 1+ \frac{(-1)^n}{\sqrt{n}} + \frac{1}{n} + o\left(\frac{1}{n}\right)\\
&=1+ \frac{(-1)^n}{\sqrt{n}} + O\left(\frac{1}{n}\right).
\end{align*}
