Find the Following Sum!

Calculus Level 2

Evaluate

limN(12N+12N+1+12N+2+12N+3++13N). \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 lnab,\ln \frac{a}{b}, where aa and bb are coprime positive integers.

Find a+ba+b.

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