Sequentia 1, 3/8, 2/9, 5/32, 2/25, ..., id est
n
+
1
2
n
2
{\displaystyle {\frac {n+1}{2n^{2}}}}
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 }}
series , quae summa exstat si sequentia summarum partium series limitem habet.
Sit
(
x
n
)
n
∈
N
∪
{
0
}
=
n
+
1
2
.
{\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }={\frac {n+1}{2}}.}
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 .}
Sequentia Fibonacci 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}}
Limes sequentiae hoc modo definitus est:
a
∈
R
{\displaystyle a\in \mathbb {R} }
(
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 }
a
{\displaystyle a}
a
:=
lim
n
→
∞
x
n
,
{\displaystyle a:=\lim _{n\to \infty }x_{n},}
a
{\displaystyle a}
convergere . Sin non est talis
a
{\displaystyle a}
divergere .
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}}\ }
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}}
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 }}
(
x
n
F
)
n
∈
N
∪
{
0
}
{\displaystyle (x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }}
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 }
divinae proportionis .
Sit
(
x
n
)
n
∈
N
=
1
n
.
{\displaystyle (x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}.}
lim
n
→
∞
1
n
=
0.
{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}=0.}
Definitio: Numerus
a
∈
R
{\displaystyle a\in \mathbb {R} }
(
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 }
Sequentiae
(
x
n
)
n
∈
N
=
1
n
{\displaystyle (x_{n})_{n\in \mathbb {N} }={\frac {1}{n}}}
Sequentiae
(
x
n
)
n
∈
N
∪
{
0
}
=
(
−
1
)
n
{\displaystyle (x_{n})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }=(-1)^{n}}
Sequentiae Fibonacci
(
x
n
F
)
n
∈
N
∪
{
0
}
{\displaystyle (x_{n}^{F})_{n\in \mathbb {N} \cup \lbrace 0\rbrace }}
Sit
(
x
n
)
n
∈
N
{\displaystyle (x_{n})_{n\in \mathbb {N} }}
a
:=
lim
n
→
∞
x
n
{\displaystyle a:=\lim _{n\to \infty }x_{n}}
Sit
(
x
n
)
n
∈
N
{\displaystyle (x_{n})_{n\in \mathbb {N} }}
a
{\displaystyle a}
(
x
n
k
)
k
∈
N
{\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }}
a
{\displaystyle a}
Si est limes
lim
n
→
∞
x
n
=
a
{\displaystyle \lim _{n\to \infty }x_{n}=a}
c
∈
R
{\displaystyle c\in \mathbb {R} \;}
lim
n
→
∞
c
⋅
x
n
=
c
⋅
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,}
lim
n
→
∞
(
c
−
x
n
)
=
c
−
a
.
{\displaystyle \lim _{n\to \infty }\left(c-x_{n}\right)=c-a.}
Si insuper
a
≠
0
{\displaystyle a\neq 0}
x
n
≠
0
{\displaystyle x_{n}\neq 0}
N
0
{\displaystyle N_{0}\;}
n
>
N
0
{\displaystyle n>N_{0}\;}
lim
n
→
∞
c
x
n
=
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}
lim
n
→
∞
y
n
=
b
{\displaystyle \lim _{n\to \infty }y_{n}=b}
lim
n
→
∞
(
x
n
+
y
n
)
=
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,}
lim
n
→
∞
(
x
n
⋅
y
n
)
=
a
⋅
b
.
{\displaystyle \lim _{n\to \infty }\left(x_{n}\cdot y_{n}\right)=a\cdot b.}
Si insuper
b
≠
0
{\displaystyle b\neq 0}
y
n
≠
0
{\displaystyle y_{n}\neq 0}
N
0
{\displaystyle N_{0}\;}
n
>
N
0
{\displaystyle n>N_{0}\;}
lim
n
→
∞
x
n
y
n
=
a
b
.
{\displaystyle \lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}.}
Nexus interni
