find-Y-n-0-dx-x-1-x-2-x-n-n-gt-1-integr-

Question Number 143261 by Mathspace last updated on 12/Jun/21

f i n d Y_{n} = \int_{0}^{\infty} \frac{d x}{(x + 1) (x + 2) \dots . (x + n)}

(n > 1 i n t e g r)

Answered by Olaf_Thorendsen last updated on 12/Jun/21

Y_{n} = \int_{0}^{1} \frac{d x}{\underset{k = 1}{\prod^{n}} (x + k)}

Y_{n} = \int_{0}^{1} \underset{k = 1}{\sum^{n}} \frac{A_{k}}{x + k} d x

A_{k} = \frac{1}{\underset{\underset{p \neq k}{p = 1}}{\prod^{n}} (- k + p)}

Y_{n} = {[\underset{k = 1}{\sum^{n}} A_{k} \ln ∣ x + k ∣]}_{0}^{1}

Y_{n} = \underset{k = 1}{\sum^{n}} A_{k} \ln (1 + \frac{1}{k})

Y_{n} = \underset{k = 1}{\sum^{n}} \frac{\ln (1 + \frac{1}{k})}{\underset{\underset{p \neq k}{k = 1}}{\prod^{n}} (p - k)}