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Long division of the series

  1. Apr 3, 2012 #1
    Not sure when this problem in my book says to calculate by long division the series 1/(1+x) = 1 - x + x^2 - x^3 + ..., and then integrating termwise between 0 and x.

    I am really rusty on these types of problems and need help understanding how to even begin T.T. Thanks for the help.
     
  2. jcsd
  3. Apr 3, 2012 #2
    I don't get the integrating between 0 and x part. But have you ever done long division of polynomials? That's all this is, one of the polynomials is [itex]p(x) = 1 + x[/itex] and the other one is [itex]q(x) = 1[/itex]. Try working with some other polynomials first, then see if you can pick up the pattern doing it this way.
     
  4. Apr 3, 2012 #3
    Not sure if I correctly implemented your response but here is what I tried:

    1/(1+x) = 1 + 1/x

    -x/(1+x) = -x - 1

    x^2/(1+x) = x^2 + x

    -x^3/(1+x) = -x^3 - x^2

    However, this seems to be incorrect because everything cancels except the 1/x

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

    I must have misinterpreted your response and this is really starting to get to me. Shouldnt everything cancel to where 1/(x+1) = 1 + 1/x ?
     
  5. Apr 3, 2012 #4
    No, you can't do this. You can't break up the denomnator like that. As a simple, example, take x = 0.


    I suggest googling "long division of polynomials" and look at some examples, then try it with your problem.
     
  6. Apr 3, 2012 #5

    HallsofIvy

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    1+ x)1- x+ x^2- x^3+ ...

    Obviously 1 divides into 1 1 time so we have then subtract
    1- x+ x^2- x^3+ ...
    1+ x
    __________________________
    -2x+ x^2- x^3+....

    And 1 divides into -2x -2x times. Multiplying and subtract
    -2x+ x^2- x^3+ ...
    -2x- 2x^2
    _________________
    3x^2- x^3+...

    Now 1 divides into that 3x^2 times so multiplying and subtracting
    3x^2- x^3+ x^4- x^5
    3x^2+3x^3
    __________________
    -3x^3+ x^4- x^5

    So far we have 1- 2x+ 3x^2. See the pattern?
     
  7. Apr 3, 2012 #6

    rcgldr

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    long hand division example:
    Code (Text):


                    1      - x + x^2 - x^3 + ...
            -------------------------------
    1 + x | 1
              1  + x
              ------
                  - x
                  - x  - x^2
                  ----------
                          x^2
                          x^2 + x^3
                          ----------
                                - x^3
                                - x^3  - x^4
                                ------------
                                            x^4
                                            ...
     
     
  8. Apr 3, 2012 #7
    Thanks. I see the pattern and realize my mistake in my previous post (so silly of me). I did it for the next few terms and got it to be 1 - 2x + 3x^2 - 4x^3 + 5x^4 - 6x^5

    How can I use the pattern to integrate termwise between 0 and x?

    I forgot to mention in the original post that we are interested in -1 < x <= 1.
     
  9. Apr 4, 2012 #8

    rcgldr

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    I think you're supposed to integrate each term, but since this is a sum, there's no reason these couldn't all be combined into one integral:

    [tex]\int_0^x 1 dx - \int_0^x x dx + \int_0^x x^2 dx - \int_0^x x^3 dx \ + \ ... [/tex]
     
    Last edited: Apr 4, 2012
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