Find-the-sum-of-x-x-1-x-x-1-x-2-x-1-x-3-for-1-1-x-lt-1-

Question Number 76794 by necxxx last updated on 30/Dec/19

F i n d t h e s u m o f

x + \frac{x}{1 + x} + \frac{x}{{(1 + x)}^{2}} + \frac{x}{{(1 + x)}^{3}} + \dots . f o r

∣ \frac{1}{1 + x} ∣< 1

Commented by turbo msup by abdo last updated on 30/Dec/19

S (x) = x (1 + \frac{1}{1 + x} + {(\frac{1}{1 + x})}^{2} + \dots .)

= \frac{x}{1 - \frac{1}{1 + x}} = \frac{x}{\frac{1 + x - 1}{1 + x}} = 1 + x

Answered by Rio Michael last updated on 30/Dec/19

S_{\infty} = \frac{x}{1 - \frac{1}{1 + x}} = \frac{x}{\frac{1 + x - 1}{1 + x}} = x . \frac{1 + x}{x} = 1 + x