Find the Following Sum!

Calculus Level 3

Evaluate

\[ \lim_{N\to\infty} \left(\frac{1}{2N}+\frac{1}{2N+1}+\frac{1}{2N+2}+\frac{1}{2N+3}+\dots+\frac{1}{3N}\right).\]

The answer is of the form \(\ln \frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers.

Find \(a+b\).

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