Results 1 to 4 of 4

Thread: Series/convergence problem

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    228

    Series/convergence problem

    Summation notation from n=1 to n=infinity

    sin(n)/n

    I believe it converges by the ratio test but my book says it's "inconclusive" because of a comparison to the harmonic series. I don't get it....... any help please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Nacho's Avatar
    Joined
    Mar 2008
    From
    Santiago, Chile
    Posts
    135
    $\displaystyle
    \sum\limits_{n \geqslant 1} {\sin \left( n \right)} < K_{ \in \mathbb{R}} {\text{ }}
    $ (you try prove it) and $\displaystyle
    \frac{1}
    {n}\xrightarrow[{n \to \infty }]{}0
    $ for dirichlet, the serie $\displaystyle
    \sum\limits_{n \geqslant 1} {\frac{{\sin \left( n \right)}}
    {n}}
    $ converge
    Last edited by Nacho; Feb 26th 2009 at 06:28 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14
    $\displaystyle \sum\limits_{n=1}^{\infty }{a_{n}\sin n}<\infty$ whenever $\displaystyle a_n$ is a decreasing sequence and $\displaystyle \lim_{n\to\infty}a_n=0.$

    Here $\displaystyle a_n=\frac1n$ and this fulfills the above conditions, whereat $\displaystyle \sum\limits_{n=1}^{\infty }{\frac{\sin n}{n}}<\infty ,$ and we're done.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,718
    Thanks
    3003
    Quote Originally Posted by Kaitosan View Post
    Summation notation from n=1 to n=infinity

    sin(n)/n

    I believe it converges by the ratio test but my book says it's "inconclusive" because of a comparison to the harmonic series. I don't get it....... any help please?
    That's very strange. You can say that a particular test for convergence is "inconclusive" but that doesn't apply to a series itself! Obviously, any given series either converges or it doesn't. That's true whether we know which is true or not!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series convergence problem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Apr 5th 2011, 07:27 AM
  2. Replies: 2
    Last Post: May 1st 2010, 09:22 PM
  3. Replies: 1
    Last Post: Mar 30th 2010, 01:44 PM
  4. Replies: 4
    Last Post: Dec 1st 2009, 03:23 PM
  5. series convergence problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 28th 2008, 12:43 PM

Search Tags


/mathhelpforum @mathhelpforum