Putting it through twice

Calculus Level 4

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

Let a=0P/2f(x)dx \displaystyle a = \int_0^{P/2} f(x) \, dx . If f(πea)f \left( \pi e^{|a|} \right) can be expressed as αβγ -\dfrac{\alpha\sqrt \beta}{\gamma}, where α,β\alpha, \beta and γ\gamma are positive integers with α,γ\alpha, \gamma coprime and β\beta square-free, find α+β+γ\alpha + \beta + \gamma.

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