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Math Help - Taylor Series using Geometric Series.

  1. #1
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    Question Taylor Series using Geometric Series.

    Let f(x) = \frac{4-4x}{4x^{2} -8x -5}; given the partial decomposition,

    \frac{4-4x}{4x^{2} -8x -5} = \frac{1}{5-2x} - \frac{1}{1+2x},

    find the Taylor series of f(x) about 1. Express your answer in sigma notation and simplify as much as possible. Dtermine the open interval of convergence.

    See figure attached for my attempt.

    Did I a mistake converting each piece into a taylor series by use of geometric series?

    Also I can't think of how to simplify this anymore, but this may be due to a mistake in the first portion of my work.

    Does anyone see any problems in my work?

    Thanks again!
    Attached Thumbnails Attached Thumbnails Taylor Series using Geometric Series.-q2a1p1.jpg   Taylor Series using Geometric Series.-q2a1p2.jpg  
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  2. #2
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    I see one problem 7 - 2(x-1) \ne 5 - 2x.
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  3. #3
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    Quote Originally Posted by Danny View Post
    I see one problem 7 - 2(x-1) \ne 5 - 2x.
    Whoops! Thank you for pointing this out. I'll fix it and reattempt the problem.

    I'll post my new results in this thread.
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  4. #4
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    Question

    Here's my 2nd crack at it!

    Things look alot better, however I want to make sure I am simplifying as much as possible. Can you spot any errors or possibly are more simplifications that can be done?

    Thanks again!
    Attached Thumbnails Attached Thumbnails Taylor Series using Geometric Series.-q2a2.jpg  
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