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Math Help - partial fractions

  1. #1
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    partial fractions

    I'm a little rusty with my partial fractions so I just need a refresher.

    The equation is 1/(x+1)(x+2)(x+4) I have
    A/x+1 + B/x+2 + C/x+4
    Then common denominators

    (x+1)(x+2)(x+4)= A(x+2)(x+4) + B(x+1)(x+4) + C(x+1)(x+2)

    After this can I do where, x=-2 for example? Cause then I will get 0=-2B
    And then keep plugging in values for x? Or is there another way which I complete forgot about
    Last edited by mr fantastic; September 20th 2010 at 07:02 PM. Reason: Corrected a typo.
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  2. #2
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    Yes there is another way.

    For starters, you should actually have gotten

    \frac{A(x + 2)(x + 4) + B(x + 1)(x + 4) + C(x + 1)(x + 2)}{(x + 1)(x + 2)(x + 4)} = \frac{1}{(x + 1)(x + 2)(x + 4)}

    A(x + 2)(x + 4) + B(x + 1)(x + 4) + C(x + 1)(x + 2)=1

    A(x^2 + 6x + 8) + B(x^2 + 5x + 4) + C(x^2 + 3x + 2) = 1

    Ax^2 + 6Ax + 8A + Bx^2 + 5Bx + 4B + Cx^2 + 3Cx + 2C = 1

    (A + B + C)x^2 + (6A + 5B + 3C)x + 8A + 4B + 2C = 0x^2 + 0x + 1.


    This means

    A + B + C = 0
    6A + 5B + 3C = 0
    8A + 4B + 2C = 1.

    Solve these equations simultaneously to find A, B, C.
    Last edited by Prove It; September 20th 2010 at 06:22 PM.
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  3. #3
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    thanks. Forgot about solving with coefficients
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  4. #4
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    so I have the series from 1 to infinity of 1/3(x+1) -1/5(x+2) +1/6(x+4)

    How do I solve the sum from there? I tried telescoping, and it doesn't work
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  5. #5
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    When you get to 1 \equiv  a(x+2)(x+4)+b(x+1)(x+4)+c(x+1)(x+2), you can put x = -1, -2, -4 to respectively find a, b, c (instead of solving the system from the coefficient relations).
    Last edited by mr fantastic; September 20th 2010 at 07:04 PM.
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  6. #6
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    Quote Originally Posted by TheCoffeeMachine View Post
    Ok. When you get to 1 \equiv  a(x+2)(x+4)+b(x+1)(x+4)+c(x+1)(x+2), you can put x = -1, -2, -4 to respectively find a, b, c (instead of solving the system from the coefficient relations).
    Will telescoping series work from there to find the sum?
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  7. #7
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    Quote Originally Posted by guyonfire89 View Post
    Will telescoping series work from there to find the sum?
    IF the series is meant to be the sum of what you posted in post #1 (which I corrected because of the typos) , then yes, it is a telescoping series. To realise this you need to:

    1. Find the correct partial fraction decomposition.

    2. Write out the first few terms ( I suggest at least the first four terms) and start cancelling in the usual way.

    If you take greater care with the details you will probably have better luck in solving questions like this one.
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  8. #8
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    Hello, guyonfire89!

    An error in your thiird statement.


    \text{The fraction is: }\;\dfrac{1}{(x+1)(x+2)(x+4)}

    \text{I have: }\;\dfrac{A}{x+1} + \dfrac{B}{x+2} + \dfrac{C}{x+4}

    We have: . \displaystyle \frac{1}{(x+1)(x+2)(x+4)} \;=\;\frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{x+4}


    Multiply through by the LCD:

    . . \displaystyle 1 \;=\;A(x+2)(x+4) + B(x+1)(x+4) + C(x+1)(x+2)

    m


    \begin{array}{cccccccccccc}<br />
\text{Let }x = \text{-}1\!: & 1 &=& A(1)(3) + B(0) + C(0) & \Rightarrow & A &=& \frac{1}{3} \\ \\[-3mm]<br />
\text{Let }x = \text{-}2\!: & 1 &=& A(0) + B(\text{-}1)(2) + C(0) & \Rightarrow & B &=& \text{-}\frac{1}{2} \\ \\[-3mm]<br />
\text{Let }x = \text{-}4\!: & 1 &=& A(0) + B(0) + C(\text{-}3)(\text{-}2) & \Rightarrow & C &=& \frac{1}{6} \end{array}


    Therefore: . \displaystyle \frac{1}{(x+1)(x+2)(x+4)} \;=\;\frac{\frac{1}{3}}{x+1} + \frac{\text{-}\frac{1}{2}}{x+2} + \frac{\frac{1}{6}}{x+4}

    . . . . . . . . . . . . . . . . . . . . . . . . \displaystyle =\;\frac{1}{6}\left(\frac{2}{x+1} - \frac{3}{x+2} + \frac{1}{x+4}\right)

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  9. #9
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    thanks guys, but when I try to prove its a telescoping series, only the first two terms cancel out. I've tried out the first 4 terms
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