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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?

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    $\begingroup$ Those numbers are $\sqrt e$ and $e^2$ $\endgroup$
    – J. W. Tanner
    Dec 28, 2022 at 22:45
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    $\begingroup$ As the others have indicated, these numbers can also be expressed in terms of $e$. +1 for your curiosity! $\endgroup$
    – Jair Taylor
    Dec 28, 2022 at 22:51
  • $\begingroup$ 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? $\endgroup$
    – ybakos
    Dec 28, 2022 at 23:04

1 Answer 1

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

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    $\begingroup$ Welcome to Mathematics Stack Exchange. Your last line has an unmatched parenthesis $\endgroup$
    – J. W. Tanner
    Dec 28, 2022 at 22:57
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    $\begingroup$ Thank you, I fixed it $\endgroup$
    – Fix
    Dec 29, 2022 at 14:44

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