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Calculus Level 3

Two infinite products AA and BB are defined as follows:

A=n=2(11n3),B=n=1(1+1n(n+1)). A = \displaystyle\prod\limits_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right), \quad B =\displaystyle\prod\limits_{n=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right).

If AB=mk,\frac{A}{B} = \frac{m}{k}, where mm and kk are relatively prime positive integers, determine 100m+k100m+k.

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