$\displaystyle e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n}$
1
$(1+1/n)^n$
$e$
$(1+1/n)^n$
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Error
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2. Series Definition
$\displaystyle e = \sum_{k=0}^{\infty}\frac{1}{k!}$
1
Partial sum $S_N = \sum_{k=0}^{N}\frac{1}{k!}$
$e$
Partial Sum $S_N$
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Error
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3. Convergence Comparison
The series $\sum 1/k!$ converges much faster than the limit $(1+1/n)^n$.
$|(1+1/n)^n - e|$ (limit)
$|S_n - e|$ (series)
4. Slope of $y = a^x$ at $x = 0$
$e$ is the unique base $a > 0$ for which the slope of $y = a^x$ at $x = 0$ is exactly $1$.
2.718
$y = a^x$
Tangent at $x = 0$ (slope $= \log a$)
Other bases
Base $a$
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Slope $= \log a$
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5. Compound Interest Interpretation
$\displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}$
With principal $P=1$, annual rate $r=100\%$, and time $t=1$ year,
the amount $A = (1+1/n)^n$ approaches $e$ as the compounding frequency $n$ increases.