3 - The Game Simplified

Lets look at what happens if our signals are not periodic. We can achieve this by setting our period T to infinity such that $\lim _{T\to \infty }\sum _{n=-\infty }^{\infty }(1/T\int _{-T/2}^{T/2}x(t^{'})e^{-j2\pi nt^{'}/T}dt^{'})e^{j2\pi nt/T},\!$
where
$\ 1/T\int _{-T/2}^{T/2}x(t^{'})e^{-j2\pi nt^{'}/T}dt^{'}=\alpha _{n}\!$
first we need to remove the restiction x(t) = x(t + T) by following these steps.
1/T $\Longrightarrow df\!$
n/T $\Longrightarrow df\!$
<math> \sum_{n=-\infty} ^\infty