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Limit in a metric space
Set
todo: clean up this definition
| context | $\langle X,d\rangle$ … metric space |
| context | $x$ … infinite seqeunce in $ X $ |
| definiendum | $\mathrm{lim}_{n\to\infty}\ x_n$ |
| range | $\varepsilon\in\mathbb R$ |
| range | $ \varepsilon>0 $ |
| range | $m\in\mathbb N$ |
| range | $m\ge 0 $ |
| range | $y\equiv\mathrm{lim}_{n\to\infty}\ x_n$ |
| postulate | $ \forall\varepsilon.\,\exists m.\,\forall (n\ge m).\,d(x_n,y)<\varepsilon $ |
Examples
$\sum_{k=1}^\infty a_n = y$
means
$ \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n a_n - y \, |<\varepsilon $
Infinite sum
$\sum_{n=0}^\text{Classical} f(n):=\lim_{m\to\infty}\sum_{n=0}^m f(n)$
Riemann integral
$h_m=\frac{(b-a)}{m}$
$\int_a^b f(x)\,{\mathrm d}x := \lim_{m\to\infty}\sum_{n=0}^{m-1} f\left(a+h_mx\right)\cdot h_m$
h = (b - a)/m;
int[f_] = Sum[f[a + h*n]*h, {n, 0, m - 1}];
(* Example f(x)=5+7x^2 *)
int[5 + 7 #^2 &] // Expand
Limit[%, m -> Infinity]
With $L_mx:=a+h_mx$, that reads
$\int_a^b f(x)\,{\mathrm d}x := (b-a)\cdot\lim_{m\to\infty} \dfrac{1}{m}\sum_{n=0}^{m-1} f\left(L_mn\right)$
Note that
$\int_a^b f(x)\,{\mathrm d}x = (b-a)\int_0^1 f(L_1(x))\,{\mathrm d}x$
so any limit of the form
$\lim_{m\to\infty} \frac{1}{m} \sum_{n=0}^{m-1} g\left(\frac{n}{m}\right)$
can be rewritten as $\int_0^1 f_g(x)\,{\mathrm d}x$, where $f_g$ is a reverse engineered function from $g$.
Infinite upper bound
$\int_a^\infty f(x)\,{\mathrm d}x := \lim_{b\to \infty} \int_a^b f(x)\,{\mathrm d}x$
Reference
Wikipedia: Limit of a seqence
Context
