I'm trying to determine what happens when R>>z for the below equation
$\frac{z\sigma}{2\varepsilon }\left ( \frac{1}{z}-\frac{1}{\sqrt{R^{2}+z^{2}}} \right )$
Like most books, which is a great annoyance, authors 'talk' about numerical approximation, asymptotic expansion, approximation technique without showing anything of sufficiency
The authors says that taylor expansion gives
$E=\frac{Q}{4\pi \varepsilon z^{2}}$
Which is utterly unhelpful and of no use for a student trying to figure the vagaries of his work.
Any help is greatly appreciated.
I think there's typo in your text, it'll be reasonable for $z>>R$ instead:
\begin{align*} \frac{z\sigma}{2\varepsilon} \left( \frac{1}{z}-\frac{1}{\sqrt{R^{2}+z^{2}}} \right) &= \frac{z\sigma}{2\varepsilon} \left[ \frac{1}{z}-\frac{1}{z}\left( 1-\frac{R^{2}}{2z^{2}} \right) \right] \\ &= \sigma \times \frac{R^{2}}{4z^{2}\varepsilon} \\ &= \frac{Q}{4\pi R^{2}} \times \frac{R^{2}}{4z^{2}\varepsilon} \end{align*}
The coefficient still not agrees, please double check all the contexts you have.