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Applying the Weierstrass M test

  1. Feb 24, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\sum_{n=1}^{\infty}\frac{1}{n^2}cosnx[/tex]

    Apply the Weierstrass M-test to the series in order to find the domain of convergence.

    2. Relevant equations



    3. The attempt at a solution

    Looking at the functions I think I have gottten to the following conclusion,

    f1(x)=cosx <= 1

    f2(x)=1/4 cos(2x) <= 1/4

    I would pick my sequence to be 1/n^2. I am sure that 1/n^2 converges but does it work as a bound on each term? And then what should I do with these two (the series and the sequence of Mk.)?
     
  2. jcsd
  3. Feb 24, 2009 #2

    Office_Shredder

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    Looks like it does

    [tex] | \frac{1}{n^2} cos(nx) | = \frac{1}{n^2} |cos(nx)| [/tex]

    and |cos(nx)| is always less than or equal to 1.
     
  4. Feb 25, 2009 #3
    That wasn't too bad. But I have a more specific question. Does the M test say that the domain of convergence is R? Since are sequence of functions is defined on R?
     
  5. Feb 25, 2009 #4

    Office_Shredder

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    Yes. The M test says the series converges uniformly on R, so in particular converges on R.
     
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