1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook