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Math Help - Series and Convergence

  1. #1
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    Series and Convergence

    Hey,

    The question ask you to find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x.

    \displaystyle\large\sum\limits^\infty_{n=0}(-1)^{n}x^{2n}

    It is alternating, so if the ratio is r = -x^2 then -x^2 must be less than 1 to converge, and if it r = x^2 then x^2 must be less than 1 to converge.

    This is where I got lost. It seems i'll have 2 different answers. How would I be able to come up with the sum of the series as a function afterward?


    Help?
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  2. #2
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    By the ratio test, x must be between -1 and 1. The endpoints x=-1 and x=1 must be checked separately. Both give divergent series by the divergence test. So the series converges if and only if -1<x<1.

    The series is geometric with first term 1, and common ratio -x^2. So the sum is \frac{1}{1+x^2}
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  3. #3
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    If you recognise that this is a geometric series with \displaystyle a = 1 and \displaystyle r = -x^2, you know that infinite geometric series only converge for \displaystyle |r| < 1.

    So \displaystyle |-x^2| < 1

    \displaystyle |x|^2 < 1

    \displaystyle |x| < 1

    \displaystyle -1 < x < 1.


    Then you don't need to resort to using the ratio test and checking endpoints.
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