Results 1 to 8 of 8
Like Tree3Thanks
  • 1 Post By romsek
  • 1 Post By SlipEternal
  • 1 Post By SlipEternal

Math Help - Can you use the Alternating Series Test on series that changes signs every 2 terms?

  1. #1
    Member
    Joined
    Oct 2011
    Posts
    75

    Can you use the Alternating Series Test on series that changes signs every 2 terms?

    The Alternating Series Test (AST) says than an alternating series \sum (-1)^n u_n = u_1 - u_2 + u_3 - u_4 + ... converges if the [tex]u_n[\tex]'s are all positive, decreasing and its limit is 0. Does that exclude using it on terms that alternate every two or three terms? Even if it switches signs reliably?

    Does this series converge or diverge?
    \sum_{n=1}^{\infty} \frac{\cos n}{n^2}.

    This series alternates sign every third term, but all the other conditions are met. If it doesn't satisfy the AST, then how would you determine it's convergence?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,333
    Thanks
    894

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Quote Originally Posted by mathDad View Post
    The Alternating Series Test (AST) says than an alternating series \sum (-1)^n u_n = u_1 - u_2 + u_3 - u_4 + ... converges if the [tex]u_n[\tex]'s are all positive, decreasing and its limit is 0. Does that exclude using it on terms that alternate every two or three terms? Even if it switches signs reliably?

    Does this series converge or diverge?
    \sum_{n=1}^{\infty} \frac{\cos n}{n^2}.

    This series alternates sign every third term, but all the other conditions are met. If it doesn't satisfy the AST, then how would you determine it's convergence?
    this isn't a good choice for an example since $\sum \dfrac 1 {n^2}$ converges

    $\left|\cos(x)\right| \leq 1$ so clearly $\sum \dfrac {\cos(x)}{n^2}$ converges

    You don't have to invoke the fact that it's an alternating series to get convergence. This series converges absolutely.
    Thanks from prasum
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2011
    Posts
    75

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    OK. How about the same series with n on the bottom: \sum_{n=1}^{\infty} \frac{\cos n}{n}.
    Last edited by mathDad; April 20th 2014 at 10:00 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,333
    Thanks
    894

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Quote Originally Posted by mathDad View Post
    OK. How about the same series with n on the bottom: \sum_{n=1}^{\infty} \frac{\cos n}{n^2}.
    You mean $\displaystyle{\sum_{n=1}^\infty} \dfrac {\cos(n)} n ?$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2011
    Posts
    75

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Yes. I posted too quick . (Will correct my post.)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,845
    Thanks
    715

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Define a sequence:

    a_n = \begin{cases}\cos 1 & \text{if }n=0 \\ \displaystyle \sum_{k=1+\left\lfloor \tfrac{\pi}{2} + 2(n-1)\pi \right\rfloor}^{\left\lfloor \tfrac{3\pi}{2} + 2(n-1)\pi \right\rfloor} \dfrac{\cos k}{k} & n \equiv 1 \pmod{2} \\ \displaystyle \sum_{k=1+\left\lfloor \tfrac{3\pi}{2} + 2(n-1)\pi \right\rfloor}^{\lfloor \tfrac{\pi}{2} + 2n\pi \right\rfloor} \dfrac{\cos k}{k} & \text{otherwise}\end{cases}

    Note that for each k \in \left[1+\left\lfloor \dfrac{\pi}{2} + 2(n-1)\pi\right\rfloor,\left\lfloor \dfrac{3\pi}{2} + 2(n-1)\pi\right\rfloor\right], \cos k < 0 and for each
    k \in \left[1+\left\lfloor \dfrac{3\pi}{2} + 2(n-1)\pi \right\rfloor, \left\lfloor \dfrac{\pi}{2} + 2n\pi \right\rfloor \right], \cos k > 0.

    Hence, \sum_{n=1}^\infty \dfrac{\cos n}{n} = \sum_{n=0}^\infty a_n = \sum_{n=0}^\infty (-1)^n |a_n|. So, you can use the alternating series test on that. You just need to show that \lim_{n \to \infty} |a_n| = 0.
    Last edited by SlipEternal; April 20th 2014 at 10:46 AM.
    Thanks from mathDad
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Oct 2011
    Posts
    75

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Quote Originally Posted by romsek View Post
    You mean $\displaystyle{\sum_{n=1}^\infty} \dfrac {\cos(n)} n ?$
    Re-add the post you just deleted. It had a useful link and you directly answered the question.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,845
    Thanks
    715

    Re: Can you use the Alternating Series Test on series that changes signs every 2 term

    Oops, I messed up a_n slightly. It should actually be:

    a_n = \begin{cases}\cos 1 & \text{if }n=0 \\ \displaystyle \sum_{k=1+\left\lfloor n\pi - \tfrac{\pi}{2}\right\rfloor}^{\left\lfloor (n+1)\pi -\tfrac{\pi}{2} \right\rfloor} \dfrac{\cos n}{n} & \text{otherwise}\end{cases}
    Thanks from mathDad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: November 4th 2012, 10:07 AM
  2. Infinite Series with Double Alternating Signs
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 9th 2011, 05:23 PM
  3. alternating series test
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 10th 2009, 10:04 PM
  4. AST alternating series test series
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 9th 2009, 09:16 AM
  5. Alternating Series Test
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 16th 2009, 01:57 PM

Search Tags


/mathhelpforum @mathhelpforum