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Math Help - Power series representation

  1. #1
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    Power series representation

    Here's the question:


    Find a power series representation for the function and determine the
    interval of convergence.


    f(x) = 1/(x + 10)
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Undefdisfigure View Post
    Here's the question:


    Find a power series representation for the function and determine the
    interval of convergence.


    f(x) = 1/(x + 10)
    we know that \frac 1{1 - x} = \sum_{n = 0}^{\infty} x^n for |x| < 1

    now note: \frac 1{x + 10} = \frac 1{10} \cdot \frac 1{(x/10) + 1} = \frac 1{10} \cdot \frac 1{1 - (-x/10)}

    now continue
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  3. #3
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    Thanks Jhevon but before I even saw your answer I found the right method in the book and solved the power series representation and found that the interval of convergence is (-10, 10).
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  4. #4
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    Here's the question I REALLY should have asked. Its a little bit more challenging.

    Find a power series representation for the function and determine the interval of convergence.


    f(x) = x/(2x^2 + 1)
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Undefdisfigure View Post
    Here's the question I REALLY should have asked. Its a little bit more challenging.

    Find a power series representation for the function and determine the interval of convergence.


    f(x) = x/(2x^2 + 1)
    well, \frac x{2x^2 + 1} = \frac d{dx} \frac 14 \ln (2x^2 + 1) = \frac 14 \frac d{dx} \ln (1 - (-2x^2))

    and you know the power series for \ln (1 - x), so just find it's derivative and plug it in...

    EDIT: in retrospect, it is perhaps easier to think of this as x times the power series of 1/(1 + 2x^2) ...oh well
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