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Homework Help: Taylors expansion question

  1. May 24, 2010 #1
    Hi, could do with some help on this question if anyone can help.
    Any help much appreciated, thanks

    Q: Suppose a function f(.) defined on the set I = {x ∈ R∣x < 1} is as follows.
    For each real number x∈ I , f(x) = 1/(1-x)
    By using the Taylor expansion of this function, show that for any real number x such that
    ∣x∣ < 1,
    f(x)= 1 + ∞
    ∑ x (to power j)
    j =1
     
  2. jcsd
  3. May 24, 2010 #2

    Cyosis

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    Hint: geometric series.
     
  4. May 24, 2010 #3
    how would i go about that then?
    Thanks
     
  5. May 24, 2010 #4

    Cyosis

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    What is the definition of a geometric series? Also what is the problem you're having with finding a Taylor series?

    Show your work!
     
  6. May 24, 2010 #5
    for a geometric series:
    S=a(k^n-1)/(k-1)

    The questions is at the top, where i have to use taylor expansion to show that for any real number...
     
  7. May 24, 2010 #6

    Cyosis

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    Yes I have read the question. It is now time for you to make an attempt and show your work in accordance with the rules of this forum.

    You can either compare it to the geometric series or find the Taylor series by differentiation. Have you tried any of that?
     
  8. May 24, 2010 #7
    would the taylor series fror 1/1-x be:

    1/1-a + x-a/(1-a)^2 +(x-a)^2/(1-a)^3 + etccc

    or should i use the mMacLaurin series


    1/1-x = 1 + x + x2 + ... + xn + ... = for |x| < 1


    thanks
     
  9. May 24, 2010 #8

    Cyosis

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    Yes find the series about the point x=0. Now how can you write that expression as an infinite series?
     
  10. May 24, 2010 #9
    how do i find the series about the point x=0?
    Do i just sub in 0 to anywhere x is in the taylor series?
     
  11. May 24, 2010 #10

    Cyosis

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    You have already done it in post #7 (also called the Maclaurin series). Now you just need to find an expression for your result using sum notation (and technically you would need to show that it converges for |x|<1).
     
  12. May 24, 2010 #11
    Okay thanks,
    do i still use the taylor series though, like i showed in 7 or not?
     
  13. May 24, 2010 #12

    Cyosis

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    That's what the question asks you to do.
     
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