All spheres are identical and initially uncharged, S₁ is closed at t=0 leaving all other switches open. After sufficient long time    is opened and   is closed. The process is done till    is opened and   is closed. Net loss of energy of system if it is made of infinite number of spheres.
(R = Radius of each sphere) 

 

 

 

 

$C=4\pi {\epsilon }_{0}R$

When Switch S₁ is closed the energy loss by battery $\frac{1}{2}C{V}^2$

when switch S2 is closed and switch S₁ is opened energy loss 

$\frac{1}{2}\frac{\mathrm{C}}{2}{\mathrm{V}}^2 \mathrm{and} \mathrm{both} \mathrm{will} \mathrm{attain} \mathrm{a} \mathrm{common} \mathrm{potential} \raisebox{1ex}{$\mathrm{V}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

when switch S₃ is closed loss of energy 

$\frac{1}{2}\frac{\mathrm{C}}{2}{\left(\frac{\mathrm{V}}{2}\right)}^2 \mathrm{and} \mathrm{their} \mathrm{common} \mathrm{potential} \mathrm{will} \mathrm{be} \frac{\mathrm{V}}{4}$

${\mathrm{\Delta U}}_{\mathrm{Cons}}=\frac{1}{2}{\mathrm{CV}}^2+\frac{1}{2}\frac{\mathrm{C}}{2}{\left(\mathrm{V}\right)}^2+\frac{1}{2}\frac{\mathrm{C}}{2}{\left(\frac{\mathrm{V}}{2}\right)}^2+\frac{1}{2}\frac{\mathrm{C}}{2}{\left(\frac{\mathrm{V}}{4}\right)}^2+........$

${\mathrm{\Delta U}}_{\mathrm{Cons}}=\frac{1}{2}{\mathrm{CV}}^2+\frac{1}{2}\frac{\mathrm{C}}{2}{\left(\mathrm{V}\right)}^2\left(1+\frac{1}{4}+\frac{1}{16}+........\right)$

${\mathrm{\Delta U}}_{\mathrm{Cons}}=\frac{5}{6}4{\mathrm{\pi \epsilon }}_0{\mathrm{RV}}^2$