Answer
$E_{\text {next }}=q d^3 / 4 \pi \varepsilon_0 z^5$.
Work Step by Step
$
\begin{aligned}
E & =\frac{q}{4 \pi \varepsilon_0 z^2}\left(\left(1+\frac{d}{z}+\frac{3}{4} \frac{d^2}{z^2}+\frac{1}{2} \frac{d^3}{z^3}+\ldots\right)-\left(1-\frac{d}{z}+\frac{3}{4} \frac{d^2}{z^2}-\frac{1}{2} \frac{d^3}{z^3}+\ldots\right)\right) \\
& =\frac{q d}{2 \pi \varepsilon_0 z^3}+\frac{q d^3}{4 \pi \varepsilon_0 z^3}+\ldots
\end{aligned}
$
Therefore, in the terminology of the problem, $E_{\text {next }}=q d^3 / 4 \pi \varepsilon_0 z^5$.
