I am exploring the basic first principles of Euler's number e. A common illustration is the compound interest rate example:
(1 + 1/n)^n
Here we describe a 100% growth rate, or "doubling," that is continuous, or "compounds really really frequently."
But what about other growth rates? For example, if the interest rate is 50%, we might calculate:
(1 + 0.5/n)^n
This number converges around 1.648.
Or, if the interest rate is 200%, we might calculate:
(1 + 2/n)^n
This number converges around 7.38.
Why aren't these numbers special, like e is?