# Derivation of $e$

It's well-known that $$e = \lim_{n\rightarrow \infty} (1+1/n)^n$$ as defined by Bernoulli when considering infinitely-compounded interest. I believe this is the earliest definition of $e$.

But if we were in (say) the 17th century (before differentiation), how would we know that the limit exists and how could we calculate the value to arbitrarily many decimal places? Equivalently, how can we prove that $$e = \sum_{n=0} 1/n!$$ without using $\frac{d}{dx}e^x = e^x$? (If we can prove $\lim_{h\rightarrow 0} \frac{e^h-1}{h} = 1$, that gives the derivative of $e^x$ and I'm fine with that approach too.)

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I am curious. Back in the 17th and 18th centuries, the Least Upper Bound Property of $\mathbb{R}$ had not been formalized yet, nor had the completeness of $\mathbb{R}$. Hence, does that mean that Euler and Bernoulli worked with the number $e$ without actually having proved that it existed? Despite the various definitions of $e$ using infinite series, infinite products, continued fractions, nested radicals, recursive formulas, etc., I feel that they were not equipped to even talk about its existence. –  Haskell Curry Dec 29 '12 at 5:55
Could you please elaborate on what "without using calculus" means? Limits and series are often considered part of "calculus". The pre-19th century version of "knowing" or "proving" something in analysis is different from what we have today. I don't understand "Equivalently" in your question. –  Jonas Meyer Dec 29 '12 at 6:19
@Jonas: I have exactly the same thoughts. Without calculus (which includes the theory of limits, infinite series, etc.), one cannot prove that the limit exists. Unless the OP defines ‘calculus’ as simply the use of derivatives and integrals, then according to such a definition, his question is well-posed. However, it is not my view that the discussion of limits should be removed from ‘calculus’. –  Haskell Curry Dec 29 '12 at 6:57
@JonasMeyer: I think the way math evolved was to use some informal versions of infinite series without any rigorous notions of limits (see for example the Madhava series, used around 1400CE, en.wikipedia.org/wiki/Madhava_series). In any case, in my question I meant "how can we derive a method to calculate digits of $e$ from the definition of Bernoulli, but without resorting to the Taylor series". –  Fixee Dec 30 '12 at 5:29

$$e = \lim_{n\rightarrow \infty} (1+1/n)^n$$

So on binomial expansion,

the 1st term is $T_0=1=\frac 1{0!},$

the $r$-th term (where integer $r\ge1$)$$T_r=\frac{n(n-1)\cdots(n-r+1)}{1\cdot2\cdots r}\frac1{n^r} =\frac1{r!}\prod_{0\le s<r}(1-\frac sn)$$

So, $$\lim_{n\rightarrow \infty}T_r=\frac1{r!}$$

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This is so minimalist that it's almost confusing. Your writing style is unique. +1 –  000 Dec 29 '12 at 6:39
Indeed, it is unique. + –  Babak S. Dec 29 '12 at 7:17
@Limitless, I'm really sorry if it sounded curt/abstruse. –  lab bhattacharjee Dec 29 '12 at 10:33
@labbhattacharjee No need to be sorry; you've got a nice style. –  000 Dec 29 '12 at 17:37

There is a paper with an excellent history by J L Coolidge, The number e, Amer. Math. Monthly 57 (1950), 591-602.

You might find the The number e on MacTutor History of Mathematics useful in exploring/answering your question.

e is so famous, it even has its own book: "e": The Story of a Number (Princeton Science Library), Eli Maor

There is also a decent Wiki History.

Note: this only answers the first part of your question as the second part is answered by someone else.

Regards

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I didn't get much out of your first link or your second link, but that book really looks promising and I do enjoy the Wiki History. +1 –  000 Dec 29 '12 at 19:07
@Limitless, I think the paper answers his first question (as good as the history gets of this number). What I like is that it shows that the founders had great difficulty getting their hands around some of the numbers we take for granted. Another excellent example is the 300-year timeline that it took for people to get their hands around the number $0$. Regards –  Amzoti Dec 29 '12 at 19:10
Great resources, Amzoti! –  amWhy May 10 '13 at 1:18