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Sum to Infinity of a Geometric Series

  1. Aug 13, 2011 #1
    1. The problem statement, all variables and given/known data

    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] + ...

    2. Relevant equations

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

    3. 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 text book: [itex]\frac{x + 1}{1 - x}[/itex]

    Can anyone help me figure out where the 2x becomes just x? Thank you.
     
  2. jcsd
  3. Aug 13, 2011 #2

    rock.freak667

    User Avatar
    Homework Helper

    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.
     
  4. Aug 13, 2011 #3
    Thanks for clearing that up. I appreciate the help.
     
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