1. Not finding help here? Sign up for a free 30min 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!

Power series representation

  1. Mar 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Use differentiation to find a power series representation for f(x) = 1/ (1+x)^2


    2. Relevant equations
    geometric series sum = 1/(1+x)



    3. The attempt at a solution
    (1) I see that the function they gave is the derivative of 1/(1+x).
    (2) Therefore, (-1)*(d/dx)summation(x^n) = -1/(1+x)^2
    (3) Differentiating the summation gives:
    (-1)*[summation (n)x^(n-1)]

    However, the book is telling me that for my second step (2) I should be getting
    d/dx [summation (-1)^n (x^n)].

    Why is it becoming an alternating series here?
     
  2. jcsd
  3. Mar 7, 2008 #2

    StatusX

    User Avatar
    Homework Helper

    Remember:

    [tex] \frac{1}{1-x} = 1+x+x^2+.... [/tex]

    so that, after substituting -x for x:

    [tex] \frac{1}{1+x} = 1 - x + x^2 + ... [/tex]

    You can remember the denominator in the first equation is 1-x by multiplying both sides by (1-x), giving:

    [tex] 1 = (1-x)(1+x+x^2+...) = 1 + x + x^2 + ... - x - x^2 -x^3 - ... = 1[/tex]

    which is consistent, as opposed to what you'd get if you assume 1+x+... was 1/(1+x). (By the way, these manipulations of infinite sums aren't strictly valid, but they can be made more rigorous by restricting to finite sums and taking a limit at the end).
     
    Last edited: Mar 7, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Power series representation
Loading...