Sum of the functions

Level pending

Let \(x, y\) be integers such that \(1 < x \leq y\).

Define \(f(x, y) = (1 - \dfrac{1}{x}) (1 - \dfrac{1}{x + 1}) (1 - \dfrac{1}{x + 2}) ... (1 - \dfrac{1}{y})\).

Let \(S = f(2, 2014) + f(3, 2014) + f(4, 2014) +...+ f(2013, 2014)\).

If \(S\) can be expressed in the form \(\dfrac{a}{b}\), where \(a, b\) are coprime, positive integers, what is the last three digits of \(a + b\)?

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