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Math Help - why do these 2 series have same interval of convergence

  1. #1
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    why do these 2 series have same interval of convergence



    \sum^{\infty}_{n=0} 1/(2+x)

    and
    \sum^{\infty}_{n=0} x^3/(x+2)


    oops they arent the series, but the power series representation so ignore the sigmas
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  2. #2
    MHF Contributor matheagle's Avatar
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    BECAUSE

    {x^3\over x+2}= x^3 \biggl({1\over 2+x}\biggr) = x^3 \biggl({1/2\over 1-(-x/2)}\biggr)

    Which is a geo with r=-x/2, which converges whenever |-x/2|<1 or -2<x<2.
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