Putting it through twice

Calculus Level 5

\[f\left( x \right) =\sum _{ n=0 }^{ \infty }{ \frac { \sin { \left( xn \right) } }{ { 7 }^{ n } } }\] 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 \(\alpha + \beta + \gamma\).

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