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Utente:Mark16/Sandbox_2

Utente:Mark16/Sandbox 2 - Teknopedia

Utente:Mark16/Sandbox 2

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${n \choose k}\cdot {n \choose n-k}^{-1}=1$ per tutti i $0\leq k\leq n$ .
$\sum _{n=1}^{m}{\frac {1}{n}}={\frac {1}{m!}}\cdot \sum _{k=0}^{m-1}{\frac {m!}{m-k}}={m \choose 1}^{-1}\cdot \sum _{k=0}^{m-1}{m \choose k}\cdot {m-1 \choose k}^{-1}$
$\sum _{n=1}^{m}{\frac {1}{n}}=\sum _{n=1}^{m}{\frac {1}{n}}\cdot {m \choose n}\cdot {m \choose m-n}^{-1}={\frac {1}{m!}}\cdot \sum _{k=0}^{m-1}{\frac {m!}{m-k}}\cdot {m \choose k}\cdot {m \choose m-k}^{-1}=$

={\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}

$\sum _{n=1}^{m}n=\sum _{k=0}^{m-1}m-k={m \choose 1}\cdot \sum _{k=0}^{m-1}{m \choose k}^{-1}\cdot {m-1 \choose k}$
$m-k={\frac {(m-k)!}{(m-k-1)!}}={m \choose 1}\cdot {m \choose k}^{-1}\cdot {m-1 \choose k}$
$\sum _{k=0}^{m-1}{m \choose k}^{-1}\cdot {m-1 \choose k}={\frac {m+1}{2}}$

Estratto da "https://it.wikipedia.org/w/index.php?title=Utente:Mark16/Sandbox_2&oldid=52424109"