\( \lim\limits_{x\to\infty} \) \( \sum\limits_{n=0}^{x}{(e-\sum\limits_{k=0}^{n}{1/k!})} \)
= \( \lim\limits_{x\to\infty} \) [(x+1)e - \( \sum\limits_{n=0}^{x}{} \) \( \sum\limits_{k=0}^{n}{1/k!} \) ] = *
Nebenrechn: ΣΣ + x*1/x! + (x-1)*1/(x-1)! + ...2*1/2! + 1*1/1! = (x+1) \( \sum\limits_{k=0}^{x}{1/k!} \) durch Umsort.
ΣΣ = (x+1) \( \sum\limits_{k=0}^{x}{1/k!} \) - \( \sum\limits_{k=0}^{x-1}{1/k!} \)
weiter: * = \( \lim\limits_{x\to\infty} \) [(x+1)e - (x+1) \( \sum\limits_{k=0}^{x}{1/k!} \) + \( \sum\limits_{k=0}^{x-1}{1/k!} \) ]
= \( \lim\limits_{x\to\infty} \) [(x+1)(e - \( \sum\limits_{k=0}^{x}{1/k!} \)) + \( \sum\limits_{k=0}^{x-1}{1/k!} \) ]
= 0 + e *)
*) wieso 0:
[(x+1)(e - \( \sum\limits_{k=0}^{x}{1/k!} \)) ]
= [(x+1)( \( \sum\limits_{k=0}^{\infty}{1/k!} \) - \( \sum\limits_{k=0}^{x}{1/k!} \)) ]
= [(x+1)( \( \sum\limits_{k=x+1}^{\infty}{1/k!} \) ) ]
= [( \( \sum\limits_{k=x+1}^{\infty}{(x+1)/k!} \) ) ]
= [1/x! + (x+1)/(x+2)! + (x+1)/(x+3)! + (x+1)/(x+4)! ... ]
= [1/x! + (x+1)/ (x+1)!(x+2) + (x+1)/ (x+2)!(x+3)! + (x+1)/ (x+3)!(x+4) ... ]
< [1/x! + (x+1)/ (x+1)!(x+1) + (x+1)/ (x+2)!(x+1) + (x+1)/ (x+3)!(x+1) ... ]
= [1/x! + 1/ (x+1)! + 1/ (x+2)! + 1/ (x+3)! ... ]
= Restglied Taylor → 0
ΣΣ = 1/0! 1/0! 1/0! ... ...1/0! 1/0!
1/1! 1/1!... 1/1! 1/1!
1/2! ... 1/2! 1/2!
1/3! 1/3!
1/(x-1)! 1/(x-1)!
1/x!
ΣΣ = 1/0! 1/0! 1/0! ... ...1/0! 1/0! hier unten drunter = Ergänzung
1/1! 1/1!... 1/1! 1/1! 1/1!
1/2! ... 1/2! 1/2! 1/2! 1/2!
1/3! 1/3! 1/3! 1/3!
1/(x-1)! ...
1/x! 1/x! ... 1/x!
Jetzt die Diagonalen zusammenzählen!
