Who's gonna be the topper from our community! RA vs SA

Calculus Level 3

Two infinite products $A$ and $B$ are defined as follows:

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

If $\frac{A}{B} = \frac{m}{k},$ where $m$ and $k$ are relatively prime positive integers, determine $100m+k$ .