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Homework Help: Convergence of a series.

  1. Feb 3, 2006 #1
    I wish I had more done.. but I'm have trouble understanding this....

    Show that the series The sum is k=0 to infinity [tex] \sum^\infty_0 1-x/(2+x)^k [/tex] converges uniformly on [0,1].

    What I have done so far is take the
    the lim is k-->infinity [tex]\lim_\infty 1-x/(2+x)^k = 0 [/tex]
    therefore the limit exists...
    does this even matter.. i mean, do I even need to state this.. because I still need to say if it converges on [0,1] which im have trouble starting on.. please help.
    thank you,
    sorry i havnt quite mastered latex
    adam
     
    Last edited: Feb 3, 2006
  2. jcsd
  3. Feb 4, 2006 #2

    arildno

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    Sloppiness with your parentheses!! :grumpy:
    (I take it you meant [itex](1-x)/(2+x)^{k}=\frac{1-x}{(2+x)^{k}}[/itex])

    Hmm.. what you have shown so for is that for any fixed x, the series does not necessarily diverge.

    As for proving uniform convergence, split it in two:

    1. Show that you have pointwise convergence.
    2. Having 1, since you have pointwise convergence on a compact set, then you also have uniform convergence on that set.
     
    Last edited: Feb 4, 2006
  4. Feb 4, 2006 #3
    this is just a test
    suppose [tex] |f_k(x)| \leq M_k [/tex]
    then, [tex] \sum^\infty_1 |f_k(x)| \leq \sum^\infty_1 M_k [/tex]
     
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