Results 1 to 2 of 2

Math Help - Convergence or divergence

  1. #1
    Member
    Joined
    May 2009
    Posts
    86

    Convergence or divergence

    I am not sure how to work this problem out.

    Investigate the behavior (convergence or divergence ) of
    ∑an if
    an = 1/(1 + z^n)
    Any help will be appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,409
    Thanks
    1294
    Quote Originally Posted by poorna View Post
    I am not sure how to work this problem out.

    Investigate the behavior (convergence or divergence ) of
    ∑an if
    an = 1/(1 + z^n)
    Any help will be appreciated
    Is z a complex number?

    If so, z = r\cos{\theta} + i\,r\sin{\theta}

    and z^n = r^n\cos{n\theta} + i\,r^n\sin{n\theta}.


    So a_n = \frac{1}{1 + z^n}

     = \frac{1}{1 + r^n\cos{n\theta} + i\,r^n\sin{n\theta}}

     = \frac{1 + r^n\cos{\theta} - i\,r^n\sin{n\theta}}{(1 + r^n\cos{n\theta})^2 + r^{2n}\sin^2{n\theta}}

     = \frac{1 + r^n\cos{\theta} - i\,r^n\sin{n\theta}}{1 + 2r^n\cos{n\theta} + r^{2n}\cos^2{\theta} + r^{2n}\sin^2{n\theta}}

     = \frac{1 + r^n\cos{\theta} - i\,r^n\sin{n\theta}}{1 +  2r^n\cos{n\theta} + r^{2n}}

     =  \frac{1 + r^n\cos{\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}} + i \left(-\frac{r^n\sin{n\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}}\right)


    So \sum_n{\left(\frac{1}{1 + z^n}\right)} = \sum_n{\left[ \frac{1 + r^n\cos{\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}} + i  \left(-\frac{r^n\sin{n\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}}\right)\right]}

     = \sum_n{\left(\frac{1 + r^n\cos{\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}}\right)} + n\,i\sum_n{\left(-\frac{r^n\sin{n\theta}}{1 +  2r^n\cos{n\theta}  + r^{2n}}\right)}.


    Now investigate the behaviour of both of these sums.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. convergence and divergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 1st 2010, 01:03 PM
  2. convergence or divergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 23rd 2009, 05:42 PM
  3. please help with convergence divergence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 15th 2009, 03:22 PM
  4. convergence and divergence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 21st 2008, 12:23 AM
  5. Convergence / Divergence
    Posted in the Calculus Forum
    Replies: 8
    Last Post: April 6th 2008, 07:51 AM

Search Tags


/mathhelpforum @mathhelpforum