For $x\neq 0$, and probably with $|x|<1$, I want to show the following equation
$$ \sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}} \right) = 1 + \sum_{n=1}^{\infty}(-1)^n(x^n+x^{n-1} + \cdots + x^{-n}) \tag{1} $$
Is this valid expression, if so how can prove this?
My simple trial is \begin{align} \sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}} \right) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{-n} - x^{n+1}}{1-x} \end{align} I tried to expand this using geometric series, but having trouble obtain R.H.S of equation (1).
Note that we have $$\begin{align} x^n+x^{n-1}+\dots+x^{-n} &=x^{-n}(1+x+\dots+x^{2n})\\ &=x^{-n}\cdot\frac{1-x^{2n+1}}{1-x}\\ &=\frac{x^{-n}-x^{n+1}}{1-x}\\ &=\frac{x^{n+1}-x^{-n}}{x-1}\\ &=\frac{x^{n+1/2}-x^{-n-1/2}}{x^{1/2}-x^{-1/2}}\\ \end{align}$$