At t = 0 , a capacitance C , a resistance R , and an emf E are...

The current in a series RC circuit as a function of time is given by:

$I(t) = \dfrac{\mathcal{E}}{R} e^{-t/RC}$

The power delivered by the battery is the product of its emf and the current flowing through the circuit:

$P(t) = \mathcal{E} I(t) = \dfrac{\mathcal{E}^2}{R} e^{-t/RC}$

Since the exponential term $e^{-t/RC}$ is a decreasing function of time for $t \ge 0$, the power delivered by the battery is maximum at the initial moment, $t = 0$.

Substituting $t = 0$ into the power equation:

P_\max = \dfrac{\mathcal{E}^2}{R} e^{0} = \dfrac{\mathcal{E}^2}{R}