Infinite number of cells having $emf$ and internal resistance $\left( {E,r} \right)$, $\left( {\frac{E}{n},\frac{r}{n}} \right)$, $\left( {\frac{E}{{{n^2}}},\frac{r}{{{n^2}}}} \right)$, $\left( {\frac{E}{{{n^3}}},\frac{r}{{{n^3}}}} \right)$..... are connected in series in same manner across an external resistance of $\frac{{nr}}{{n + 1}}$ . Current flowing through the external resistor is
Diffcult
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$I=\frac{E+\frac{E}{n}+\frac{E}{n^{2}}+\frac{E}{n^{3}} \ldots \ldots}{\left(r+\frac{r}{n}+\frac{r}{n^{2}}+\frac{r}{n^{3}}+\ldots\right)+\frac{n r}{n+1}}$
$\frac{{E\left[ {\frac{1}{{1 - \frac{1}{n}}}} \right]}}{{r\left[ {\frac{1}{{1 - \frac{1}{n}}}} \right] + \frac{{nr}}{{n + 1}}}}$
$\mathrm{I}=\frac{\frac{\mathrm{nE}}{(\mathrm{n}-1)}}{\frac{\mathrm{m}}{\mathrm{n}-1}+\frac{\mathrm{nr}}{\mathrm{n}+1}}=\mathrm{I}=\frac{\frac{\mathrm{E}}{(\mathrm{n}-1)}}{\left(\frac{\mathrm{n}+1+\mathrm{n}-1}{(\mathrm{n}-1)(\mathrm{n}+1)}\right) \mathrm{r}}$
$=\frac{(n+1) E}{2 n r}$
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