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How does $$\sqrt{R^2 + |x|^2} = R + \frac{|x|^2}{2R}+\cdots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out.

Michael Hardy
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user35687
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    Note, what is small is $\frac{|x|}{R} \ll 1$. Therefor: $\sqrt{R^2+|x|^2}=R\sqrt{1+\left(\frac{x}{R}\right)^2}\approx R\left(1+\frac 1 2 \frac{x^2}{R^2}\right)$ – Ali Sep 11 '14 at 20:52

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$$ f(w)=\sqrt{R^2+w}. \qquad f(0) = R. $$ $$ f'(w) = \frac{1}{2\sqrt{R^2+w}}. \qquad f'(0)=\frac1{2R}. $$ $$ \text{For small }w,\quad f(w) \approx f(0) + f'(0)w = R + \frac w{2R}. $$

Michael Hardy
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