Sequentia (mathematica)

Sequentia 1, 3/8, 2/9, 5/32, 2/25, ..., id est ${\frac {n+1}{2n^{2}}}$${\frac {n+1}{2n^{2}}}$ Limis sequentiae est 0.

Sequentia in mathematica omnis quidem functio $f:\mathbb {N} \cup \lbrace 0\rbrace \rightarrow \mathbb {R} ,\ n\mapsto f(n)=:(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$$f:\mathbb {N} \cup \lbrace 0\rbrace \rightarrow \mathbb {R} ,\ n\mapsto f(n)=:(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$ appellatur. Summa membrorum sequentiae cuiusdam est series, quae summa exstat si sequentia summarum partium series limitem habet.

Exempla

• Sit $(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.$$(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.$ Sequentia numerorum tum est ${\frac {1}{2}},{\frac {2}{2}}=1,{\frac {3}{2}},{\frac {4}{2}}=2,\dots .$${\frac {1}{2}},{\frac {2}{2}}=1,{\frac {3}{2}},{\frac {4}{2}}=2,\dots .$
• Sequentia Fibonacci: Sequentia Fibonacci est sequentia recursive definita. (Id est: numeri principales sequentiae positi sunt et formula ad numerum proximum numeris positis putandum data est).

$x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}$$x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}$. Ergo sequentia est: 0,1,1,2,3,5,8,13,21,... .

Limes et puncta auctus sequentiae

Limes sequentiae

Limes sequentiae hoc modo definitus est:

$a\in \mathbb {R}$$a\in \mathbb {R}$ est limes sequentiae $(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }:\Longleftrightarrow \forall \varepsilon >0\,\exists N_{\varepsilon }\in \mathbb {N} \,\forall n\geq N_{\varepsilon }\,:\left|x_{n}-a\right|<\varepsilon$$(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }:\Longleftrightarrow \forall \varepsilon >0\,\exists N_{\varepsilon }\in \mathbb {N} \,\forall n\geq N_{\varepsilon }\,:\left|x_{n}-a\right|<\varepsilon$. Si sequentiae est limes $a$$a$, scribitur: $a:=\lim _{n\to \infty }x_{n},$$a:=\lim _{n\to \infty }x_{n},$ et sequentia dicitur ad $a$$a$ convergere. Sin non est talis $a$$a$, sequentia dicitur divergere.

Exempla

• Sequentiae superiori scriptae $(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}\$$(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}\$ et $\ x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}$$\ x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}$ divergunt.
• Sequentia autem $\left({\frac {x_{n-1}^{F}}{x_{n}^{F}}}\right)_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$$\left({\frac {x_{n-1}^{F}}{x_{n}^{F}}}\right)_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$, ubi $(x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$$(x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$ sit sequentia Fibonacci, convergit et limes est $\lim _{n\to \infty }{\frac {x_{n-1}^{F}}{x_{n}^{F}}}={\frac {1+{\sqrt {5}}}{2}}=:\phi$$\lim _{n\to \infty }{\frac {x_{n-1}^{F}}{x_{n}^{F}}}={\frac {1+{\sqrt {5}}}{2}}=:\phi$ numerus divinae proportionis.
• Sit $(x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}.$$(x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}.$ Tum $\lim _{n\to \infty }{\frac {1}{n}}=0.$$\lim _{n\to \infty }{\frac {1}{n}}=0.$

Puncta auctus sequentiae

Definitio: Numerus $a\in \mathbb {R}$$a\in \mathbb {R}$ est punctum auctus sequentiae $(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }:\Longleftrightarrow \forall \varepsilon >0\,\forall n\in \mathbb {N} \,\exists m\geq n:\left|x_{m}-a\right|<\varepsilon$$(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }:\Longleftrightarrow \forall \varepsilon >0\,\forall n\in \mathbb {N} \,\exists m\geq n:\left|x_{m}-a\right|<\varepsilon$

Exempla

• Sequentiae $(x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}$$(x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}$ est punctum auctus 0.
• Sequentiae $(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }=(-1)^{n}$$(x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }=(-1)^{n}$ sunt puncta auctus et 1 et -1.
• Sequentiae Fibonacci $(x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$$(x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }$ non est punctum auctus.

Cohaerentia limitis punctorumque auctus sequentiae

1. Sit $(x_{n})_{n\in \mathbb {N} }$$(x_{n})_{n\in \mathbb {N} }$ sequentia aliqua convergens et $a:=\lim _{n\to \infty }x_{n}$$a:=\lim _{n\to \infty }x_{n}$ sit eius limes. Tum a est punctum auctus.
2. Sit $(x_{n})_{n\in \mathbb {N} }$$(x_{n})_{n\in \mathbb {N} }$ sequentia aliqua quae punctum auctus $a$$a$ habet. Tum est sequentia partitiva $(x_{n_{k}})_{k\in \mathbb {N} }$$(x_{n_{k}})_{k\in \mathbb {N} }$, quae habet punctum auctus $a$$a$ limitem.

Theoremata limitum

Si est limes $\lim _{n\to \infty }x_{n}=a$$\lim _{n\to \infty }x_{n}=a$, tum omni numero $c\in \mathbb {R} \;$$c\in \mathbb {R} \;$ sunt limites hi, qui eo modo putentur:

• $\lim _{n\to \infty }c\cdot x_{n}=c\cdot a,$$\lim _{n\to \infty }c\cdot x_{n}=c\cdot a,$
• $\lim _{n\to \infty }\left(c+x_{n}\right)=c+a,$$\lim _{n\to \infty }\left(c+x_{n}\right)=c+a,$
• $\lim _{n\to \infty }\left(c-x_{n}\right)=c-a.$$\lim _{n\to \infty }\left(c-x_{n}\right)=c-a.$

Si insuper $a\neq 0$$a\neq 0$ est, tum etiam $x_{n}\neq 0$$x_{n}\neq 0$ a quodam numero indicabili $N_{0}\;$$N_{0}\;$ et sequentiae partitivae $n>N_{0}\;$$n>N_{0}\;$ valet:

$\lim _{n\to \infty }{\frac {c}{x_{n}}}={\frac {c}{a}}.$$\lim _{n\to \infty }{\frac {c}{x_{n}}}={\frac {c}{a}}.$

Si sunt limites et $\lim _{n\to \infty }x_{n}=a$$\lim _{n\to \infty }x_{n}=a$ et $\lim _{n\to \infty }y_{n}=b$$\lim _{n\to \infty }y_{n}=b$, tum etiam limites hi sunt, qui eo modo putentur:

• $\lim _{n\to \infty }\left(x_{n}+y_{n}\right)=a+b,$$\lim _{n\to \infty }\left(x_{n}+y_{n}\right)=a+b,$
• $\lim _{n\to \infty }\left(x_{n}-y_{n}\right)=a-b,$$\lim _{n\to \infty }\left(x_{n}-y_{n}\right)=a-b,$
• $\lim _{n\to \infty }\left(x_{n}\cdot y_{n}\right)=a\cdot b.$$\lim _{n\to \infty }\left(x_{n}\cdot y_{n}\right)=a\cdot b.$

Si insuper $b\neq 0$$b\neq 0$ est, tum etiam $y_{n}\neq 0$$y_{n}\neq 0$ a quodam numero indicabili $N_{0}\;$$N_{0}\;$ et sequentiae partitivae $n>N_{0}\;$$n>N_{0}\;$ valet:

$\lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}.$$\lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}.$

Nexus interni

• Fibonacci
• Divina proportio
• Limes functionis