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!

Need help understanding the binomial series

  1. Apr 24, 2014 #1
    1. The problem statement, all variables and given/known data
    My math textbook is currently on the Binomial Series now, after completing the Binomial Theorem (no problems with that one). I believe most of my trouble comes from the book's rather glancing explanation of it, only giving examples of the form ##(1 +/- kx)##. Now have encountered this question:

    Q
    Obtain the first three terms, in ascending powers of x, of the expansion of ##(8+3x)^\frac{2}{3}##, stating the set of values for which this expansion is valid.


    2. Relevant equations

    The binomial series is defined as :

    ##(1+x)^n = 1 + nx + \frac{n(n+1)}{1*2}x^2 + \frac{n(n+1)(n-2)}{1*2*3}x^3 + ... ##, provided that |x| < 1.

    Expansion of ##(1+kx)^n## is valid for ##\frac{-1}{k} < x < \frac{1}{k}##

    3. The attempt at a solution

    I am able to find the expansion by following the binomial series definition, but multiplying everything by ##8^n##, where n is 2/3, -1/3, -4/3, and so on.

    ##(8+3x)^\frac{2}{3} = 8^\frac{2}{3} + (8^\frac{-1}{3})(\frac{2}{3})(3x) + (8^\frac{-4}{3})\frac{(\frac{2}{3})(\frac{-1}{3})}{2}(3x^2)##

    Which is simplified to :

    ## 4 + x - \frac{x^2}{16} ## This is correct, according to the textbook answers.

    However, I cannot find the range of x for which it is valid. The textbook answer is :
    ##\frac{-8}{3} < x < \frac{8}{3}##.

    Why is that so? Could I have an explanation?
     
  2. jcsd
  3. Apr 24, 2014 #2

    jedishrfu

    Staff: Mentor

    If you pull the 8 out of the expression (8 + 3x) = 8*(1 + (3/8)x ) does that help?
     
  4. Apr 24, 2014 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    If you write [tex](8+3x)^n = 8^n \left(1 + \frac{3}{8}x \right)^n[/tex] the reason should become clear.

    Also: I hope you realize that the restrictions on ##x## apply only if ##n## is not a positive integer. When ##n## is a positive integer (n = 1 or 2 or 3 or .... ) the expansion is valid for all values of ##x##.
     
  5. Apr 24, 2014 #4
    D'oh. Thanks guys :P
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Need help understanding the binomial series
  1. Binomial Series (Replies: 5)

  2. Binomial series (Replies: 3)

  3. Binomial Series (Replies: 5)

Loading...