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Convergent series. Is my logic correct?

  1. Dec 7, 2008 #1
    Show [tex]
    \sum_1^\infty\frac{x^n}{1+x^n} [/tex] converges when x is in [0,1)
    [tex]
    \sum_1^\infty\frac{x^n}{1+x^n} = \sum_1^\infty\frac{1}{1+x^n} * x^n <= \sum_1^\infty\frac{1}{1} * x^n = \sum_1^\infty x^n [/tex]

    The last sum is g-series, converges since r = x < 1
     
  2. jcsd
  3. Dec 8, 2008 #2

    HallsofIvy

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    Yes, that is correct. In fact, you can say more: the series converges for x in (-1, 1).
     
  4. Dec 8, 2008 #3

    mathwonk

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    ratio test?
     
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