Inverse summation

Calculus Level 4

If \(\displaystyle \sum_{n=1}^{\infty}\cot^{-1}\left(\frac{2}{1729}+\frac{(n+n^{2})1729}{2} \right)=\tan^{-1}\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, then what is the value of \(a+b\)?

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