A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in mind, I want to know if
$$\frac{1}{\sqrt{a}}=1-\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n-1)(2n)!!}\left ( 1-\frac{1}{a} \right )^n$$ where $a \geq 1$ is true or not.
First, this series can be simplified to $\sum_{n=1}^{\infty}\frac{(2n-3)!!}{(2n)!!}(1-\frac{1}{a})^n$. Dunno if it helps or not.
Second, I used the divergence test, and it does not give me any information on whether it converges or diverges. If I can show that it diverges, then the equality is obviously false. But this didn't happen.
Third, I can attempt to write $\frac{1}{\sqrt{a}}$ as $(\sqrt{a}^{-1})$. Perhaps I can convert this to the form of $(1+x)^n$, and then proceed with the binomial theorem?