# Infinite dielectric environment: total charge of polarization proof

Consider an infinite environment with electrical permittivity non-homogeneous $\epsilon=\epsilon_0(1+a/r)$ a being a positive constant. A conducting sphere of radius R and charge Q is put on that environment, centered at r=0. Determine the electric field $E$, the electrical potential $V$, the volume density of polarization charge $\rho'$ and the surface density of polarization charge $\sigma'$. Prove that the total charge of polarization in the dielectric is zero.

So I'm having a lot of trouble with the last part of the question. I got for the the volume density of polarization charge $\rho'$ and the surface density of polarization charge $\sigma'$:

$\rho'= \frac{Qa}{4\pi r^2(r+a)^2}$ for $r>R$

$\sigma'= \frac{-Qa}{4\pi R^2(R+a)}$ for $r=R$

Now I now that the total charge will be the sum of the total charge in surface with the total charge in volume. Total charge in surface is easy. $Q'_{surface}=4\pi R^2\sigma'= \frac{-Qa}{(R+a)}$

However I'm not quite sure on how to compute the total volume density. I thought it might be an integral from R to infinity of

$\frac{4}{3} \pi r^3 \frac{Qa}{4 \pi r^2 (r+a)^2}=\frac{Qar}{3(r+a)^2}$

but the integral of that expression on those boundaries diverge. What should I do then? Any suggestions?