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Sequentia (mathematica)

Sequentia (mathematica)

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Sequentia (mathematica)
ExemplaLimes et puncta auctus sequentiaeLimes sequentiaeExemplaPuncta auctus sequentiaeExemplaCohaerentia limitis punctorumque auctus sequentiaeTheoremata limitumNexus interni

Sequentia in mathematica omnis quidem functio f : N ∪ { 0 } → R ,   n ↦ f ( n ) =: ( x n ) n ∈ N ∪ { 0 } {\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 }} {\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.

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Sequentia 1, 3/8, 2/9, 5/32, 2/25, ..., id est n + 1 2 n 2 {\displaystyle {\frac {n+1}{2n^{2}}}} {\displaystyle {\frac {n+1}{2n^{2}}}} Limis sequentiae est 0.
  • Sit ( x n ) n ∈ N ∪ { 0 } = n + 1 2 . {\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.} {\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.} Sequentia numerorum tum est 1 2 , 2 2 = 1 , 3 2 , 4 2 = 2 , … . {\displaystyle {\frac {1}{2}},{\frac {2}{2}}=1,{\frac {3}{2}},{\frac {4}{2}}=2,\dots .} {\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).
x 0 := 0 ,   x 1 := 1 ,   x n := x n − 2 + x n − 1 {\displaystyle x_{0}:=0,\ x_{1}:=1,\ x_{n}:=x_{n-2}+x_{n-1}} {\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 sequentiae

Limes sequentiae hoc modo definitus est:

a ∈ R {\displaystyle a\in \mathbb {R} } {\displaystyle a\in \mathbb {R} } est limes sequentiae ( x n ) n ∈ N ∪ { 0 } :⟺ ∀ ε > 0 ∃ N ε ∈ N ∀ n ≥ N ε : | x n − a | < ε {\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 } {\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 a {\displaystyle a} {\displaystyle a}, scribitur: a := lim n → ∞ x n , {\displaystyle a:=\lim _{n\to \infty }x_{n},} {\displaystyle a:=\lim _{n\to \infty }x_{n},} et sequentia dicitur ad a {\displaystyle a} {\displaystyle a} convergere. Sin non est talis a {\displaystyle a} {\displaystyle a}, sequentia dicitur divergere.

Exempla

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

Puncta auctus sequentiae

Definitio: Numerus a ∈ R {\displaystyle a\in \mathbb {R} } {\displaystyle a\in \mathbb {R} } est punctum auctus sequentiae ( x n ) n ∈ N ∪ { 0 } :⟺ ∀ ε > 0 ∀ n ∈ N ∃ m ≥ n : | x m − a | < ε {\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 } {\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 ( x n ) n ∈ N = 1 n {\displaystyle (x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}} {\displaystyle (x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}} est punctum auctus 0.
  • Sequentiae ( x n ) n ∈ N ∪ { 0 } = ( − 1 ) n {\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }=(-1)^{n}} {\displaystyle (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 ∈ N ∪ { 0 } {\displaystyle (x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }} {\displaystyle (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 ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} {\displaystyle (x_{n})_{n\in \mathbb {N} }} sequentia aliqua convergens et a := lim n → ∞ x n {\displaystyle a:=\lim _{n\to \infty }x_{n}} {\displaystyle a:=\lim _{n\to \infty }x_{n}} sit eius limes. Tum a est punctum auctus.
  2. Sit ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} {\displaystyle (x_{n})_{n\in \mathbb {N} }} sequentia aliqua quae punctum auctus a {\displaystyle a} {\displaystyle a} habet. Tum est sequentia partitiva ( x n k ) k ∈ N {\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }} {\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }}, quae habet punctum auctus a {\displaystyle a} {\displaystyle a} limitem.

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

  • lim n → ∞ c ⋅ x n = c ⋅ a , {\displaystyle \lim _{n\to \infty }c\cdot x_{n}=c\cdot a,} {\displaystyle \lim _{n\to \infty }c\cdot x_{n}=c\cdot a,}
  • lim n → ∞ ( c + x n ) = c + 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,}
  • lim n → ∞ ( c − x n ) = c − 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 a ≠ 0 {\displaystyle a\neq 0} {\displaystyle a\neq 0} est, tum etiam x n ≠ 0 {\displaystyle x_{n}\neq 0} {\displaystyle x_{n}\neq 0} a quodam numero indicabili N 0 {\displaystyle N_{0}\;} {\displaystyle N_{0}\;} et sequentiae partitivae n > N 0 {\displaystyle n>N_{0}\;} {\displaystyle n>N_{0}\;} valet:
lim n → ∞ c x n = c a . {\displaystyle \lim _{n\to \infty }{\frac {c}{x_{n}}}={\frac {c}{a}}.} {\displaystyle \lim _{n\to \infty }{\frac {c}{x_{n}}}={\frac {c}{a}}.}

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

  • lim n → ∞ ( x n + y n ) = a + b , {\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,}
  • lim n → ∞ ( x n − y n ) = a − b , {\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,}
  • lim n → ∞ ( x n ⋅ y n ) = a ⋅ b . {\displaystyle \lim _{n\to \infty }\left(x_{n}\cdot y_{n}\right)=a\cdot b.} {\displaystyle \lim _{n\to \infty }\left(x_{n}\cdot y_{n}\right)=a\cdot b.}
Si insuper b ≠ 0 {\displaystyle b\neq 0} {\displaystyle b\neq 0} est, tum etiam y n ≠ 0 {\displaystyle y_{n}\neq 0} {\displaystyle y_{n}\neq 0} a quodam numero indicabili N 0 {\displaystyle N_{0}\;} {\displaystyle N_{0}\;} et sequentiae partitivae n > N 0 {\displaystyle n>N_{0}\;} {\displaystyle n>N_{0}\;} valet:
lim n → ∞ x n y n = a b . {\displaystyle \lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}.} {\displaystyle \lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}.}
  • Fibonacci
  • Divina proportio
  • Limes functionis
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Exempla

Limes et puncta auctus sequentiae

Theoremata limitum

Nexus interni