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Range of validity for Binomial Series

  1. Oct 11, 2014 #1
    1. The problem statement, all variables and given/known data

    Dear Mentors and Helpers,
    here's the question:
    Find the range of validity for (1 + 3x/2)^(-1) and (1 + 1/(3x))^(-1).
    2. Relevant equations


    3. The attempt at a solution
    For the first binomial series:

    -1 < 3x/2 < 1
    -2 < 3x < 2 (multiply 2 throughout)
    -2/3 < x < 2/3 (divide by 3 throughout)

    For the second binomial series:
    -1 < 1/(3x) < 1
    -3x < 1 < 3x (multiply 3x throughout)

    -3x < 1 or 1 < 3x
    x > -1/3 (divide -3, inequality change sign) or 1/3 < x (divdide 3)

    Therefore I get: x > -1/3 or x > 1/3

    However my range isn't correct for the second binomial series can any Mentors or PF helper guide me and correct my mistakes.

    Thank you
     
  2. jcsd
  3. Oct 11, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    For the second series you multiply each part of the inequality by 3x. When you multiply by -3, you note that it is negative so you change the direction of the inequality- but 3x might be negative also. So you must also consider the sign of 3x.

    If 0< 1/3x< 1 then 3x is positive so 0< 1< 3x, x> 1/3. If -1< 1/3x< 0 then 3x is negative so -3x> 1> 0 and x<-1/3, not ">".
     
  4. Oct 11, 2014 #3
    Thank you for the explanation, I finally understand :w
     
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