===== Limit in a metric space ===== ==== Set ==== >todo: clean up this definition | @#55CCEE: context | @#55CCEE: $\langle X,d\rangle$ ... metric space | | @#55CCEE: context | @#55CCEE: $x$ ... infinite seqeunce in $ X $ | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}_{n\to\infty}\ x_n$ | | @#DDDDDD: range | @#DDDDDD: $\varepsilon\in\mathbb R$ | | @#DDDDDD: range | @#DDDDDD: $ \varepsilon>0 $ | | @#DDDDDD: range | @#DDDDDD: $m\in\mathbb N$ | | @#DDDDDD: range | @#DDDDDD: $m\ge 0 $ | | @#DDDDDD: range | @#DDDDDD: $y\equiv\mathrm{lim}_{n\to\infty}\ x_n$ | | @#55EE55: postulate | @#55EE55: $ \forall\varepsilon.\,\exists m.\,\forall (n\ge m).\,d(x_n,y)<\varepsilon $ | ----- === Examples === $\sum_{k=1}^\infty a_k = y$ means $ \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n a_k - 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 == $\int_a^\infty f(x)\,{\mathrm d}x := \lim_{b\to \infty} \int_a^b f(x)\,{\mathrm d}x$ === sci === >>7944543 I take one for the notebook: In calculus/analysis, infinity isn't used as an entity (like a number), but instead >limit n to infinity means >for whatever m you choose (arbitry), there is such and such, that such and such For example you may consider a sequence [math] s_n [/math] given by 1/2, 1/4, 1/8, 1/16, ... then the "limit of n to infinity" is the number y=0. Why? Because for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that [math] s_n [/math] became smaller than [math] \varepsilon [/math]. For example, choose the small number [math] \varepsilon = 0.0003 [/math]. The sequence actualy becomes forever smaller than that after, say, m = 4000. Indeed, for any n after 4000, the number [math] s_n [/math] is something smaller than 1/4000, which is 0.00025. So the limit to infinity is formalized as something to do with >for arbitrary high values of the index, the thing itself is still restricted. Coming back to the definition: The limit of n to infinity of a seqeunce [math] s_n [/math] is y, if for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that the difference (here given by the distance on the real number line) between the value [math] s_n [/math] and this y became smaller than [math] \varepsilon [/math]. In formulas [math] \lim_{n \to \infty} s_n = y[/math] iff [math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) . \, \exists ( m \in {\mathbb N} ) . \, \forall ( n \ge_{\mathbb N} m) . \, | s_n - y \, |<\varepsilon [/math] Another example: Consider again the sequence [math] s_n [/math] given by 1/2, 1/4, 1/8, 1/16, ... and create a new sequence [math] S_m = \sum_{k=1}^n a_n [/math] which has members 1/2, 1/2+1/4, 1/2+1/4+1/8, ... You can prove that with y=1 the above formula regarding [math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) [/math] holds. So we say [math] \sum_{k=0}^\infty s_n := \lim_{n \to \infty} \sum_{k=0}^n s_k = 1 [/math] === Reference === Wikipedia: [[http://en.wikipedia.org/wiki/Limit_of_a_sequence|Limit of a seqence]] ----- === Context === [[Metric space]], [[Infinite sequence]]