let-x-be-a-posative-real-number-prove-that-n-1-n-1-x-1-x-n-1-x-

Question Number 115910 by Eric002 last updated on 29/Sep/20

l e t x b e a p o s a t i v e r e a l n u m b e r

p r o v e t h a t

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

Commented by Dwaipayan Shikari last updated on 29/Sep/20

\frac{1}{(x + 1)} + \frac{1!}{(x + 1) (x + 2)} + \frac{2!}{(x + 1) (x + 2) (x + 3)} + \dots .

take x = 1

\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \dots = \underset{n = 1}{\sum^{\infty}} \frac{1}{n (n + 1)} = \frac{1}{1}

Which is true