Sum to Infinity of a Geometric Series

[odolwa99]

Homework Statement

Q. Find, in terms of x, the sum to infinity of the series...

1 + $(\frac{2x}{x + 1})$ + $(\frac{2x}{x + 1})^2$ + ...

Homework Equations

S$\infty$ = $\frac{a}{1 - r}$

The Attempt at a Solution

S$\infty$ = $\frac{a}{1 - r}$

a = 1

r = U2/ U1 = $(\frac{2x}{x + 1})$/ 1 = $\frac{2x}{x + 1}$

$\frac{1}{1 - (2x/ x + 1)}$

$\frac{x + 1}{1 - 2x}$

Ans.: From textbook: $\frac{x + 1}{1 - x}$

Can anyone help me figure out where the 2x becomes just x? Thank you.

 

[rock.freak667]

If you take

$$\frac{1}{x-\frac{2x}{x+1}} \times \frac{x+1}{x+1}$$

you will get

$$\frac{x+1}{1(x+1)-2x}$$

Which simplifies to the answer you want.

 

[odolwa99]

Thanks for clearing that up. I appreciate the help.