Classical Limits

Calculus Level 3

An unknown function \(f(x)\) can be approximated by \(g_n(x)\),

where \(g_n(x) = \frac{1^{x}+2^{x}+3^{x}+...+n^{x}}{n(1^{x-1}+2^{x-1}+3^{x-1}+...+n^{x-1})}\).

If the \(\lim_{n\to\infty}g_n(x) = f(x)\).

What is \(\prod_{i=1}^{999} f(i)\)

This problem is not original.

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