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?
They are in fact special numbers. Notice that because $$\lim_{n\to \infty}(1+\frac{1}{n})^n=e$$ we can explore$$\lim_{n\to \infty}(1+\frac{a}{n})^n=\lim_{x\to \infty}(1+\frac{1}{x})^{x\cdot a}$$ with $$\frac{a}{n}=\frac{1}{x}$$ and it follows with exponent rules and because the inner limit converges $$\left(\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x\right)^a=e^a$$