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f(x)=∑n=0∞sin(xn)7nf\left( x \right) =\sum _{ n=0 }^{ \infty }{ \frac { \sin { \left( xn \right) } }{ { 7 }^{ n } } }f(x)=n=0∑∞7nsin(xn) For all real numbers xxx, let f(x)f(x) f(x) be a function with fundamental period PP P.
Let a=∫0P/2f(x) dx \displaystyle a = \int_0^{P/2} f(x) \, dx a=∫0P/2f(x)dx. If f(πe∣a∣)f \left( \pi e^{|a|} \right) f(πe∣a∣) can be expressed as −αβγ -\dfrac{\alpha\sqrt \beta}{\gamma}−γαβ, where α,β\alpha, \betaα,β and γ\gammaγ are positive integers with α,γ\alpha, \gammaα,γ coprime and β\betaβ square-free, find α+β+γ\alpha + \beta + \gammaα+β+γ.
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