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I was reading an article that approximated a square root operator as follows

$\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $

At first glance that looks like a Taylor series expansion, but I can't replicate the result. A two term Taylor series expansion about the origin with respect to $y$ is

$f(0)+\frac{f'(0)}{1!}y+O(y^2)$

Simplifying results in

$=\sqrt{1+x}+\frac{1}{2}(1+x)^{-1/2}y+O(y^2)$

Am I doing something incorrectly or is the approximation not quite a Taylor series?

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  • $\begingroup$ It looks like a multivariable Taylor expansion. $\endgroup$ – user122283 Mar 28 '14 at 1:12
  • $\begingroup$ ... except that $\sqrt{1+x}$ is not a polynomial. $\endgroup$ – Robert Israel Mar 28 '14 at 1:39
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$(1+x)^{-1/2} = 1 + O(x)$, so your second term $\frac{1}{2} (1+x)^{-1/2} y = \frac{y}{2} + O(xy)$.

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