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lim-n-k-1-n-1-k-1-k-n-1-C-k-1-n-C-r-n-r-n-r-

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Question Number 136854 by Raxreedoroid last updated on 26/Mar/21
lim_(n→∞) Σ_(k=1) ^n ((−1)^(k+1) k×^(n−1) C_(k−1) )=? ^n C_r =((n!)/(r!(n−r)!))
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {k}×\:^{{n}−\mathrm{1}} {C}_{{k}−\mathrm{1}} \right)=? \\ $$$$\:^{{n}} {C}_{{r}} =\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$
Commented by mr W last updated on 27/Mar/21
Σ_(k=1) ^n (−1)^(k+1) kC_(k−1) ^(n−1) =0 for any n≥3
$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} =\mathrm{0}\:{for}\:{any}\:{n}\geqslant\mathrm{3} \\ $$
Answered by mr W last updated on 27/Mar/21
(1−x)^n =Σ_(k=0) ^n C_k ^n (−x)^k (1−x)^n =Σ_(k=0) ^n (−1)^k (n/k)C_(k−1) ^(n−1) x^k −n(1−x)^(n−1) =Σ_(k=0) ^n (−1)^k nC_(k−1) ^(n−1) x^(k−1) (1−x)^(n−1) =Σ_(k=0) ^n (−1)^(k+1) C_(k−1) ^(n−1) x^(k−1) x(1−x)^(n−1) =Σ_(k=0) ^n (−1)^(k+1) C_(k−1) ^(n−1) x^k (1−x)^(n−1) −x(n−1)(1−x)^(n−2) =Σ_(k=0) ^n (−1)^(k+1) kC_(k−1) ^(n−1) x^(k−1) (1−x)^(n−1) −x(n−1)(1−x)^(n−2) =Σ_(k=1) ^n (−1)^(k+1) kC_(k−1) ^(n−1) x^(k−1) let x=1 0=Σ_(k=1) ^n (−1)^(k+1) kC_(k−1) ^(n−1) lim_(n→∞) Σ_(k=1) ^n (−1)^(k+1) kC_(k−1) ^(n−1) =0
$$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{k}} ^{{n}} \left(−{x}\right)^{{k}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \frac{{n}}{{k}}{C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}} \\ $$$$−{n}\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {nC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$${x}\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} −{x}\left({n}−\mathrm{1}\right)\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{2}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} −{x}\left({n}−\mathrm{1}\right)\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{2}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$${let}\:{x}=\mathrm{1} \\ $$$$\mathrm{0}=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} =\mathrm{0} \\ $$

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