Putting it through twice

Calculus Level 4

$f\left( x \right) =\sum _{ n=0 }^{ \infty }{ \frac { \sin { \left( xn \right) } }{ { 7 }^{ n } } }$ For all real numbers $x$ , let $f(x)$ be a function with fundamental period $P$ .

Let $\displaystyle a = \int_0^{P/2} f(x) \, dx$ . If $f \left( \pi e^{|a|} \right)$ can be expressed as $-\dfrac{\alpha\sqrt \beta}{\gamma}$ , where $\alpha, \beta$ and $\gamma$ are positive integers with $\alpha, \gamma$ coprime and $\beta$ square-free, find $\alpha + \beta + \gamma$ .