Sum to Infinity of a Geometric Series

[odolwa99]

Homework Statement

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

1 + [itex](\frac{2x}{x + 1})[/itex] + [itex](\frac{2x}{x + 1})^2[/itex] + ...

Homework Equations

S[itex]\infty[/itex] = [itex]\frac{a}{1 - r}[/itex]

The Attempt at a Solution

S[itex]\infty[/itex] = [itex]\frac{a}{1 - r}[/itex]

a = 1

r = U2/ U1 = [itex](\frac{2x}{x + 1})[/itex]/ 1 = [itex]\frac{2x}{x + 1}[/itex]

[itex]\frac{1}{1 - (2x/ x + 1)}[/itex]

[itex]\frac{x + 1}{1 - 2x}[/itex]

Ans.: From textbook: [itex]\frac{x + 1}{1 - x}[/itex]

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

 

[rock.freak667]

If you take

[tex]\frac{1}{x-\frac{2x}{x+1}} \times \frac{x+1}{x+1}[/tex]

you will get

[tex]\frac{x+1}{1(x+1)-2x}[/tex]

Which simplifies to the answer you want.

 

[odolwa99]

Thanks for clearing that up. I appreciate the help.