Sequentia (mathematica)

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

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

Exempla

• Sit ${\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.}$ Sequentia numerorum tum est ${\displaystyle {\frac {1}{2}},{\frac {2}{2}}=1,{\frac {3}{2}},{\frac {4}{2}}=2,\dots .}$
• 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).

${\displaystyle x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}}$. 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:

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

Exempla

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

Puncta auctus sequentiae

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

Exempla

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

Cohaerentia limitis punctorumque auctus sequentiae

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

Theoremata limitum

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

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

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

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

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

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

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

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

Nexus interni

• Fibonacci
• Divina proportio
• Limes functionis