Pagina principale
Una pagina a caso
Nelle vicinanze
entra
Impostazioni
Fai una donazione
Informazioni su Wikipedia
Avvertenze
Ricerca
Utente
:
Mark16/Sandbox 2
Lingua
Segui
Modifica
<
Utente:Mark16
(
n
k
)
⋅
(
n
n
−
k
)
−
1
=
1
{\displaystyle {n \choose k}\cdot {n \choose n-k}^{-1}=1}
per tutti i
0
≤
k
≤
n
{\displaystyle 0\leq k\leq n}
.
∑
n
=
1
m
1
n
=
1
m
!
⋅
∑
k
=
0
m
−
1
m
!
m
−
k
=
(
m
1
)
−
1
⋅
∑
k
=
0
m
−
1
(
m
k
)
⋅
(
m
−
1
k
)
−
1
{\displaystyle \sum _{n=1}^{m}{\frac {1}{n}}={\frac {1}{m!}}\cdot \sum _{k=0}^{m-1}{\frac {m!}{m-k}}={m \choose 1}^{-1}\cdot \sum _{k=0}^{m-1}{m \choose k}\cdot {m-1 \choose k}^{-1}}
∑
n
=
1
m
1
n
=
∑
n
=
1
m
1
n
⋅
(
m
n
)
⋅
(
m
m
−
n
)
−
1
=
1
m
!
⋅
∑
k
=
0
m
−
1
m
!
m
−
k
⋅
(
m
k
)
⋅
(
m
m
−
k
)
−
1
=
{\displaystyle \sum _{n=1}^{m}{\frac {1}{n}}=\sum _{n=1}^{m}{\frac {1}{n}}\cdot {m \choose n}\cdot {m \choose m-n}^{-1}={\frac {1}{m!}}\cdot \sum _{k=0}^{m-1}{\frac {m!}{m-k}}\cdot {m \choose k}\cdot {m \choose m-k}^{-1}=}
=
1
m
!
⋅
∑
k
=
0
m
−
1
(
m
k
)
⋅
(
m
−
1
k
)
−
1
⋅
(
m
−
1
)
!
=
(
m
1
)
−
1
⋅
∑
k
=
0
m
−
1
(
m
k
)
⋅
(
m
−
1
k
)
−
1
{\displaystyle ={\frac {1}{m!}}\cdot \sum _{k=0}^{m-1}{m \choose k}\cdot {m-1 \choose k}^{-1}\cdot (m-1)!={m \choose 1}^{-1}\cdot \sum _{k=0}^{m-1}{m \choose k}\cdot {m-1 \choose k}^{-1}}
∑
n
=
1
m
n
=
∑
k
=
0
m
−
1
m
−
k
=
(
m
1
)
⋅
∑
k
=
0
m
−
1
(
m
k
)
−
1
⋅
(
m
−
1
k
)
{\displaystyle \sum _{n=1}^{m}n=\sum _{k=0}^{m-1}m-k={m \choose 1}\cdot \sum _{k=0}^{m-1}{m \choose k}^{-1}\cdot {m-1 \choose k}}
m
−
k
=
(
m
−
k
)
!
(
m
−
k
−
1
)
!
=
(
m
1
)
⋅
(
m
k
)
−
1
⋅
(
m
−
1
k
)
{\displaystyle m-k={\frac {(m-k)!}{(m-k-1)!}}={m \choose 1}\cdot {m \choose k}^{-1}\cdot {m-1 \choose k}}
∑
k
=
0
m
−
1
(
m
k
)
−
1
⋅
(
m
−
1
k
)
=
m
+
1
2
{\displaystyle \sum _{k=0}^{m-1}{m \choose k}^{-1}\cdot {m-1 \choose k}={\frac {m+1}{2}}}
