Interesting limits resulting in exp

exp_limits.tex

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\subsection{Binomial theorem}

$\left(x+y\right)^n =\sum_{k=0}^n \{\frac{n!}{k!\,(n-k)!}\}\, x^k y^{n-k} = x^n + x\,y\,\sum_{k=1}^{n-1}\{ \frac{n!}{k!\,(n-k)!} \}x^{k-1} y^{n-(k+1)} + y^n$

e.g.

$(x + y)^2 = x^2 + \{2\}xy + y^2$

$(x+y)^3 = x^3 + \{3\}x^2y +\{ 3\}xy^2 + y^3$

$(x+y)^4 = x^4 + \{4\}x^3y + \{6\}x^2y^2 + \{4\}xy^3 + y^4$

$...$

and so

$\left(1+y\right)^n = \sum_{k=0}^n\{ \frac{n!}{(n-k)!}\} \{\frac {1} {k!} \}y^k$

\subsection{Exponential series}

${\rm exp}(x)^y = {\rm exp}(x\cdot y)$

where

${\rm exp}(x) := \lim_{N\to\infty}\sum_{k=0}^N\{  \frac{1}{k!}\}x^k = 1 + x + x^2\left( \{\frac{1}{2!}\}+\{\frac{1}{3!}\}x+\{\frac{1}{4!}\}x^2+ \dots \right)$

${\rm exp}(x) =  ({\rm e}^{\frac{x}{n}})^n = \left(1+\frac{x}{n} + \left(\frac{x}{n} \right)^2\lim_{N\to\infty}\sum_{k=0}^N\{ \frac{1}{(k+2)!}\}\left(\frac{x}{n} \right)^{k}\right)^n  = \left(1+\frac{x}{n} \right)^n +\,O(\left(\frac{x}{n} \right)^2)$

Coefficient comparison:

$\left(1 + \frac {x} {n} \right)^n = \sum_{k=0}^n\{ \frac {n!} {(n-k)!\,n^k} \}  \{\frac {1} {k!} \} x^k = \sum_{k=0}^n\{\prod_{j=1}^{k}(1-\frac{k-j}{n})\} \{\frac {1} {k!} \} x^k$

Also

${\rm exp}(x) = \lim_{N\to\infty}\left(1+\frac{x}{N}+\,O((\frac{1}{N})^2)\right)^N$, \ \ \ \ e.g. $ {\rm exp}(x) =  \lim_{N\to\infty} \left(\frac{N}{N-x}\right)^N$