# A Hypothetical RC Circuit

$R(a) = \sum_{n=1}^{\infty} \frac{1}{a^n} = \frac{1}{a-1} \qquad \qquad C(a) = \sum_{n=0}^{\infty} \frac{1}{a^n} = \frac{a}{a-1}$

An RC ciruit consisting of a multitude of resistors ($$R$$) in series followed by a multitude of capacitors ($$C$$) in parallel is prepared such that the resistance and capacitance can be calculated with the series written above. At $$a= 3$$, calculate the voltage ($$V$$) of the capacitor when it has charged for $$t = 2.5 \space \text{seconds}$$, where the circuit's power supply voltage ($$V_{\circ}$$) is $$14 \space \text{volts}$$.

Note: $$V(t) = V_{\circ}(1-e^{N})$$ where $$N = \frac{\text{-t}}{\text{RC}}$$.