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Thread: Does this series converge?

  1. #1
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    Does this series converge?

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

    Well, so far I have this:

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

    So the series converge point wise to $\displaystyle \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 $\displaystyle \sum ^ \infty _{k=1} x^k $ on (0,1)? Pointwise or Uniformly?

    Well, so far I have this:

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

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

    Well, so far I have this:

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

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

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