Question-212762

Question Number 212762 by universe last updated on 23/Oct/24

Commented by MrGaster last updated on 23/Oct/24

Rewrite the integrand function as :

(x - 1) (x - 2) \dots (x - n) = \frac{{(- 1)}^{n}}{n!} x (x - 1) (x - 2) \dots (x - n)

Computational integral :

\int_{0}^{n} (x - 1) - (x - 2) \dots (x - n) d x = \int_{0}^{n} \frac{{(- 1)}^{n}}{n!} x (x - 1) (x - 2) \dots (x - n) d x

take notice of, x (x - 1) (x - 2) \dots (x - n)

Is a descending factorial polynomial . Tayi

x = 0, 1, 2, \dots n The value is 0 . So the integral is divided into two parts . :

\int_{0}^{n} (x - 1) (x - 2) \dots (x - n) d x = \int_{0}^{1} (x - 1) (x - 2) \dots (x - n) d x \int_{1}^{2} (x - 1) (x - 2) \dots (x - n) d x + \dots + \int_{n + 1}^{n} (x - 1) (x - 2) \dots (x - n) d x

Between each cell [k, k + 1], function

(x - 1) (x - 2) \dots (x - k) (x - k) (x - k) (- (k + 1)) \dots (x - n)

The sign of is the same so the absolutel

vaue can be extracted and the integralt

beween each cell can be calculated .

because [k, k + 1] Symbol invariance can be written as :

\int_{k}^{k + 1} ∣ x - k ∣ x - (k + 1) ∣ \dots ∣ x - n ∣ d x

Because the integration result betweenc

eah cell will contain a factor k!

When divided by n! when

These integral results will be eliminatedo

s the whole integral result is 0 .

\lim_{n \to \infty} \frac{{(- 1)}^{n}}{n!} \times 0 = 0

Answered by mathmax last updated on 23/Oct/24

p (x) = (x - 1) (x - 2) \dots . . (x - n) e s t u n p o l y n o m e d e d e g r e n

\Rightarrow p (x) = x^{n} + {a x}^{n - 1} + {b x}^{n - 2} + \dots \dots + l \Rightarrow

\int_{0}^{n} p (x) d x = {[\frac{x^{n + 1}}{n + 1} + \frac{{a x}^{n}}{n} + \dots . . + l x]}_{0}^{n}

= \frac{n^{n + 1}}{n + 1} + \frac{{a n}^{n}}{n} + \dots . . + l n \Rightarrow

\frac{{(- 1)}^{n}}{n!} \int_{0}^{n} p (x) d x = \frac{{(- 1)}^{n}}{n!} {\frac{n^{n + 1}}{n + 1} + \frac{{a n}^{n}}{n} + \dots . + l n}

e t ∣ \frac{{(- 1)}^{n}}{n!} \int_{0}^{n} p (x) d x ∣⩽ \frac{1}{n!} \frac{n^{n + 1}}{n + 1} + ∣ a ∣ \frac{n^{n}}{n! n}

+ \dots . . ∣ l ∣ \frac{n}{n!} \to 0 (n \to \infty)

c e s t l a f a c t o r i e l l e q u i c o m m m a n d e \Rightarrow

{l i m}_{n \to \infty} \frac{{(- 1)}^{n}}{n!} \int_{0}^{n} (x - 1) (x - 2) \dots (x - n) d x = 0

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