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Math Help - Does this series converge?

  1. #1
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    Does this series converge?

    Does the series  \sum ^ \infty _{k=1} x^k on (0,1)? Pointwise or Uniformly?

    Well, so far I have this:

    Define the partial sum S_n = \sum ^n _{k=1} x^k =1+x=x^2+...+x^n= \frac {1-x^{n+1}}{1-x}

    So the series converge point wise to  \frac {1}{1-x} ?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Does the series  \sum ^ \infty _{k=1} x^k on (0,1)? Pointwise or Uniformly?

    Well, so far I have this:

    Define the partial sum S_n = \sum ^n _{k=1} x^k =1+x=x^2+...+x^n= \frac {1-x^{n+1}}{1-x}

    So the series converge point wise to  \frac {1}{1-x} ?
    Just consider \lim_{x\to{0^-}}\sum_{n=0}^{\infty}x^n and use abel's theorem and then you can see that it does not converge unifomrly because it does not equal \frac{1}{1-0}
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    Does the series  \sum ^ \infty _{k=1} x^k on (0,1)? Pointwise or Uniformly?

    Well, so far I have this:

    Define the partial sum S_n = \sum ^n _{k=1} x^k =1+x+x^2+...+x^n= \frac {1-x^{n+1}}{1-x}

    So the series converge point wise to  \frac {1}{1-x} ?
    The sum starts at k=1, not k=0, so you need to modify the formula for S_n slightly.

    To see whether or not the series converges uniformly, you need to think about what happens as x\to1.
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