Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

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

$$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{k}} ^{{n}} \left(−{x}\right)^{{k}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \frac{{n}}{{k}}{C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}} \\ $$$$−{n}\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {nC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$${x}\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {C}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} −{x}\left({n}−\mathrm{1}\right)\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{2}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{1}} −{x}\left({n}−\mathrm{1}\right)\left(\mathrm{1}−{x}\right)^{{n}−\mathrm{2}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} {x}^{{k}−\mathrm{1}} \\ $$$${let}\:{x}=\mathrm{1} \\ $$$$\mathrm{0}=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {kC}_{{k}−\mathrm{1}} ^{{n}−\mathrm{1}} =\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com