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Sum to Infinity of a Geometric Series

  1. Aug 14, 2011 #1
    1. The problem statement, all variables and given/known data

    Q. Find the range of values of x for which the sum to infinity exists for each of these series:

    (i) 1 + [itex]\frac{1}{x}[/itex] + [itex]\frac{1}{x^2}[/itex] + [itex]\frac{1}{x^3}[/itex] + ...

    (ii) [itex]\frac{1}{3}[/itex] + [itex]\frac{2x}{9}[/itex] + [itex]\frac{4x^2}{27}[/itex] + [itex]\frac{8x^3}{81}[/itex] + ...

    2. Relevant equations

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

    3. The attempt at a solution

    (i) r = [itex]\frac{1}{x}[/itex]/ 1 = [itex]\frac{1}{x}[/itex] [itex]\Rightarrow[/itex] 1 = x
    Ans.: From text book: IxI > 1

    (ii) r = [itex]\frac{2x}{9}[/itex]/ [itex]\frac{1}{3}[/itex] = [itex]\frac{6x}{9}[/itex] [itex]\Rightarrow[/itex] 6x = 9 [itex]\Rightarrow[/itex] x = [itex]\frac{9}{6}[/itex] [itex]\Rightarrow[/itex] x = [itex]\frac{3}{2}[/itex]

    Ans.: From text book: -[itex]\frac{3}{2}[/itex] < x < [itex]\frac{3}{2}[/itex]

    I'm confused as to whether I'm approaching this correctly, or if I've simply gone wrong in expressing the answers I found. Can someone help me figure this out? Thanks.
  2. jcsd
  3. Aug 14, 2011 #2


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    Homework Helper

    For your sum to infinity to exist

    Sn must converge as r→∞.

    i.e. for |r| < 1

    so in your first one, you correctly found r as r = 1/x so it would converge for |1/x| < 1 and you know that |X|< A ⇒ -A<X<A.
  4. Aug 14, 2011 #3
    Ok, I think I see it now. Thanks for clearing that up.
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