Is there a closed form to the equation L = $(\frac{x+a}{x})^n -\left (\frac{x-a}{x}\right)^n$

Question

I have the following equation to resolve:

$$L = \left(\dfrac{x+a}{x}\right)^n - \left(\dfrac{x-a}{x}\right)^n$$

I've found a similar equation on this question, I couldn't understand line 2 to line 3 : $F_Y(\theta+\epsilon) - F_Y(\theta-\epsilon) = 1-\left(\dfrac{\theta-\epsilon}{\theta}\right)^n$? I believe understanding this I'm able to replicate in my equation.

Answer 1

I'm not quite sure that this is what you want, but this is one of the closed forms.

$$(1+a)^n-(1-a)^n=\sum_{k=0}^{n}\binom{n}{k}a^k-\sum_{k=0}^{n}\binom{n}{k}(-a)^k$$ $$=\sum_{k=0}^{n}\binom{n}{k}\{a^k-(-a)^k\}=2\sum_{k=1}^{\lfloor\frac n2\rfloor}\binom{n}{2k-1}a^{2k-1}.$$