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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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    @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, http://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".2012-12-30

2 Answers 2

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

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

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

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    @labbhattacharjee No need to be sorry; you've got a nice style.2012-12-29
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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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    Great resources, Amzoti!2013-05-10