This is a pretty basic question, but I'm just wondering if my equation is correct. We're all familiar with the standard compound interest formula,
$$P(t) = P_0 e^{rt}$$
Where $P_0$ is the initial amount of money, and $P(t)$ is the amount of money given an annual interest rate of $r$ over $t$ years. I'm wondering how this equation would change if the money represented by $P_0$ was earned linearly over $t$ years.
I thought the equation to represent this would be
$$ lim_{n \to \infty} \frac{P_0}{n} \sum_{i=1}^n e^{rTi/n} $$
The intuition behind this equation is that it is splitting up $P_0$ into $n$ parts, and similarly splitting up the time interval $t$ into $n$ parts. Each part of $P_0$ would experience a different amount of time under compound interest.
I reduced this down to
$$ \frac{(e^{rT} - 1) P_0}{rT} $$
Is my original equation as the limit, and therefore my final equation, correct?