# Computing square root of rank-1 correction

In the post Computing square root of diag(u)-uu'?, both @YaroslavBulatov and @NickAlger suggested the factor $$\frac{\sqrt{u^{T}D^{-1}u+1}-1}{u^{T}D^{-1}u}$$ for their approximation.

Can somebody explain the source of this factor, which is different from the ordinary $$\frac{1}{2}\choose{n}$$ of the power expansion.