# Other e-like numbers based on different rates of growth?

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?

• Those numbers are $\sqrt e$ and $e^2$ Dec 28, 2022 at 22:45
• As the others have indicated, these numbers can also be expressed in terms of $e$. +1 for your curiosity! Dec 28, 2022 at 22:51
• Aha, I see that the 50% rate is really e^(1/2), and the 200% rate is e^2. Can anyone post a simple algebraic proof that shows e^rate = (1+ rate/n)^n with some exponent juggling? Dec 28, 2022 at 23:04

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 $$\left(\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x\right)^a=e^a$$