# 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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