lim_(n→∞) Σ_(k=1) ^n ((−1)^(k+1) k×^(n−1) C_(k−1) )=? ^n C


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Question Number 136854 by Raxreedoroid last updated on 26/Mar/21

$\lim_{n \to \infty} \underset{k = 1}{\sum^{n}} ({(- 1)}^{k + 1} k \times^{n - 1} C_{k - 1}) = ?$ $^{n} C_{r} = \frac{n!}{r! (n - r)!}$
Commented by mr W last updated on 27/Mar/21

$\underset{k = 1}{\sum^{n}} {(- 1)}^{k + 1} {k C}_{k - 1}^{n - 1} = 0 f o r a n y n ⩾ 3$
	Answered by mr W last updated on 27/Mar/21

	${(1 - x)}^{n} = \underset{k = 0}{\sum^{n}} C_{k}^{n} {(- x)}^{k}$ ${(1 - x)}^{n} = \underset{k = 0}{\sum^{n}} {(- 1)}^{k} \frac{n}{k} C_{k - 1}^{n - 1} x^{k}$ $- n {(1 - x)}^{n - 1} = \underset{k = 0}{\sum^{n}} {(- 1)}^{k} {n C}_{k - 1}^{n - 1} x^{k - 1}$ ${(1 - x)}^{n - 1} = \underset{k = 0}{\sum^{n}} {(- 1)}^{k + 1} C_{k - 1}^{n - 1} x^{k - 1}$ $x {(1 - x)}^{n - 1} = \underset{k = 0}{\sum^{n}} {(- 1)}^{k + 1} C_{k - 1}^{n - 1} x^{k}$ ${(1 - x)}^{n - 1} - x (n - 1) {(1 - x)}^{n - 2} = \underset{k = 0}{\sum^{n}} {(- 1)}^{k + 1} {k C}_{k - 1}^{n - 1} x^{k - 1}$ ${(1 - x)}^{n - 1} - x (n - 1) {(1 - x)}^{n - 2} = \underset{k = 1}{\sum^{n}} {(- 1)}^{k + 1} {k C}_{k - 1}^{n - 1} x^{k - 1}$ $l e t x = 1$ $0 = \underset{k = 1}{\sum^{n}} {(- 1)}^{k + 1} {k C}_{k - 1}^{n - 1}$ $\lim_{n \to \infty} \underset{k = 1}{\sum^{n}} {(- 1)}^{k + 1} {k C}_{k - 1}^{n - 1} = 0$
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