Results 1 to 4 of 4

Math Help - prove that this series converges

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    prove that this series converges

    \sum \frac{e^{in\theta}}{\sqrt{n}}\\
    using Abel law
    ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The 'Abel's test' for the convergence of a series is described here...

    Abel's Uniform Convergence Test -- from Wolfram MathWorld

    In the case You have proposed we can observe that...

    a) the sequence e^{in\theta} is not 'monotonic decreasing'...

    b) the series \sum_{n} \frac{1}{\sqrt{n}} diverges...

    It seems that the result of the Abel's test in this case is negative...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    i tried to go by that link to get a conclusion
    1.
    u_n(x) can be written u_n(x)=a_nf_n(x),

    this expession is too weird for this law
    <br />
e^{in\theta}<br />

    i dont know how to apply it here
    ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The series you have proposed is the 'powers series'...

    \sum_{n=1}^{\infty} \frac{z^{n}}{\sqrt{n}} (1)

    ... computed in z=e^{i\theta} , i.e. on the circle with center in z=0 and radious 1. It is easy to demonstrate that (1) converges for |z|<1, i.e. for all points internal to that circle. A general criterion to extablish what happens on the circle of convergence doesn't exist. In this case we can say that the (1) diverges for \theta=0... for other points a detailed study is required...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove this series converges
    Posted in the Calculus Forum
    Replies: 5
    Last Post: August 18th 2011, 10:55 AM
  2. How do i prove a series converges?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 2nd 2010, 06:35 PM
  3. Prove if series converges or diverges
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 3rd 2010, 05:29 PM
  4. Replies: 2
    Last Post: September 10th 2009, 07:04 PM
  5. Prove that the series 1/n^2 converges
    Posted in the Calculus Forum
    Replies: 7
    Last Post: March 27th 2008, 07:42 PM

Search Tags


/mathhelpforum @mathhelpforum