For electric potential I have the following infinite series:
$V=k_e\frac{q}{r}+2k_eq\sum_{n=1}^{\infty}\frac{1}{\sqrt{r^2+n^2a^2}}$
Taking the derivative with respect to r, I have the following infinite series for electric field:
$E_r=k_eq\frac{1}{r^2}+k_eq\sum_{n=1}^{\infty}(r^2+n^2a^2)^{-3/2}2r$
I am trying to approximate the sum as an integral to show that, as $r$ gets very large, the electric field approaches the field due to a line charge $E=2k_e\frac{\lambda}{r}$, but I am having trouble getting this result. What I get is:
$E_r=k_eq\frac{1}{r^2}+k_eq\lim_{t\rightarrow\infty}\int_{1}^{t}(r^2+r^2a^2)^{-3/2}2rdr$
ignoring the first term because it will go away when taking the limit as $r$ gets large,
$E_r=2(1+a^2)k_eq\lim_{t\rightarrow\infty}\int_{1}^{t}r^{-2}dr$
so,
$E_r=2(1+a^2)^{-3/2}k_eq$.
I keep getting this result and am not sure where my mistake is. Any suggestions?