Fluctuating series!

Calculus Level 2

\[\frac 1{503}, \ \frac 4{524}, \ \frac 9{581}, \ \frac {16}{692}, \ \ldots \]

The above is the sequence \(\left\{u_n = \frac {n^2}{500 + 3n^3}\right\} \) with \(n=1, 2, 3, 4, \ldots.\)

If the largest term of this sequence can be expressed as \(\dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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