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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.

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