An Infinite Summation Involving Sinc Function

Calculus Level 5

$\sum_{n=1}^\infty \dfrac{1}{n}\int_{2\pi n} ^\infty \dfrac{\sin{z}}{z}dz = \pi(A-\log{\sqrt{B\pi}})$

If the equation above holds true for positive integers $$A$$ and $$B$$, then find $$A+B$$.

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