Compound Interest vs Continuous Interest I still do not fully grasp the concepts of compound and continuous interest. I understand that continuous interest is for things that cannot be expressed in years, and it is usually applied in nature. However, continuous interest is interest over a set period of time.
Here is the continuous interest formula: 
$$A = P * e^{rt}$$
Here is the compound interest formula: 
$$A = P\left( 1 + \frac{r}{n}\right)^{nt}$$
Note: $A$ is amount, $P$ is principal, $r$ is rate, $n$ is times compounded each year, and $t$ is number of years.
I am still confused, because if I have compound interest every month ($n = 12$), it would be the same as if I had continuous interest!
Can someone help clarify the difference between these two formulae?
 A: The simplest, most obvious difference is that continuous compounding uses $e$ in its function. Continuous compounding will generate the most interest of any type of compounding because of this. 
As @Anonymous noted, as you increase the number of times you compound in the discrete compounding case, you will get closer and closer to the continuous compounding formula.
Realistically, you will never come across continuous compounding in your personal investing--I've only seen it be used between banks.
A: Continuous interest is a form of compound interest. It is compounded continuously, where the period of compounding is infinitely small.
So even if the period of compounding is per second, it would still be compounded, but not continuously. 
See: http://www.investopedia.com/ask/answers/050115/what-difference-between-continuous-compounding-and-discrete-compounding.asp
hope this helps! 
A: From a mathematician's viewpoint,
The sequence $(1+\frac{r}{n})^{nt}$ is a strictly monotone increasing function of $n$. So the more compounding cycles in a fixed duration with fixed rate will yield more interest. The above expression has a limit (equivalent to continuous compounding) as $n \to \infty$. The limit is not $\infty$ but $e^{t}$.
A: Compound Interest is calculated at finite intervals when interest is added to the principal $P$:
$$ \frac{A}{P} =  ( 1 + r/n )^{nt}  $$
Let us say the interest is calculated not annually, not half-yearly, not quarterly, not monthly , weekly but every second of the duration when  in debt. ( $n \to \infty$ money lending not too generous ..)
Then P can be calculated mathematically using the definition of exponential function bringing in the base $e$ into finance usefully as: 
$$ \frac{A}{P} = e^{rt}. $$
It can be seen however that weekly compounded interest $ + P = A$ is not significantly more even if computed every milli-second, by plotting the above function.
