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Math Help - power series of rep

  1. #1
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    power series of rep

    Sum (-1)^n/(n 2^n) x^n

    0. Find the interval of convergence of the power series defining f(x).

    0. Find a power seies for f'(x) and determine its interval of convergence.

    0. Find a power seies for integral 0 to x, f(t) dt and determine its interval of convergence.

    any help would be appreciated!
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  2. #2
    MHF Contributor matheagle's Avatar
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    The ratio test gives you -1<x/2<1

    giving you (-2,2), now check the end points.
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  3. #3
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by twofortwo View Post
    Sum (-1)^n/(n 2^n) x^n

    0. Find the interval of convergence of the power series defining f(x).

    0. Find a power seies for f'(x) and determine its interval of convergence.

    0. Find a power seies for integral 0 to x, f(t) dt and determine its interval of convergence.

    any help would be appreciated!

    Set up the Ratio Test.

    |\frac{a_{n+1}}{a_n}|=|x| \frac{n}{2(n+1)}

    which approaches

    \frac{|x|}{2} as n tends to infinity

    Set this less than 1, then determine the radius and interval of convergence.

    As to parts b) and c), there is a theorem that says that the radius of convergence remains the same for both the derivative and the antiderivative.

    Good luck!
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  4. #4
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    thank you. but how do i find the power series for the integral and derivative? i can use the theorem afterwards for the intervals but how do i go about finding the power series of derivitive and integral?
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  5. #5
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by apcalculus View Post
    Set up the Ratio Test.

    |\frac{a_{n+1}}{a_n}|=|x| \frac{n}{2(n+1)}

    which approaches

    \frac{|x|}{2} as n tends to infinity

    Set this less than 1, then determine the radius and interval of convergence.

    As to parts b) and c), there is a theorem that says that the radius of convergence remains the same for both the derivative and the antiderivative.

    Good luck!
    The interval remains the same, but you can gain an endpoint or two when
    you integrate. Likewise you can lose endpoint when you differentiate.
    So you need to check the endpoints after integrating and differentiating.
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