Can I use a calculator?

Geometry Level 3

$\begin{aligned} A &=& \tan^{-1} \left( \frac 1 {\sqrt2} \right)+ \tan^{-1} \left( \frac 1 {\sqrt3} \right) + \tan^{-1} \left( \frac 1 {\sqrt4} \right) \\ \\ B & = & \frac {\pi}{2} \\ \end{aligned}$

Without resorting to tables or calculator, determine which of these numbers above is larger.

All credit for this problem goes to Pi Han Goh.