f-x-x-1-x-2-x-n-1-calculate-f-x-n-1-2-decompose-F-1-f-

Question Number 174469 by Mathspace last updated on 01/Aug/22

f (x) = (x + 1) (x + 2) \dots . (x + n)

1) c a l c u l a t e f^{^{'}} (x) (n ⩾ 1)

2) d e c o m p o s e F = \frac{1}{f}

Answered by Ar Brandon last updated on 01/Aug/22

y = (x + 1) (x + 2) \dots (x + n)

lny = \ln (x + 1) + \ln (x + 2) + \cdot \cdot \cdot + \ln (x + n)

\frac{1}{y} \cdot \frac{d y}{d x} = \frac{1}{x + 1} + \frac{1}{x + 2} + \cdot \cdot \cdot + \frac{1}{x + n}

\Rightarrow \frac{d y}{d x} = \underset{k = 1}{\prod^{n}} (x + k) \underset{k = 1}{\sum^{n}} \frac{1}{x + k}

\frac{1}{(x + 1) (x + 2) \dots (x + n)}

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

= \underset{k = 1}{\sum^{n}} \frac{{(- 1)}^{k - 1}}{(n - k)! (k - 1)! (x + k)}