Results 1 to 7 of 7

Math Help - Summation of e^(-k n^2)

  1. #1
    Newbie
    Joined
    Aug 2012
    From
    India
    Posts
    6

    Summation of e^(-k n^2)

    Hi All,

    This is my first thread so please forgive any mistakes regarding forum rules.

    I have been trying to do some calculations for my Physics project. And I get stuck at following summations,

    1. \sum_{n=-\infty}^{\infty}\exp{\left(-\alpha n^2\right)}

    2. \sum_{n=-\infty}^{\infty}n^2\exp{\left(-\alpha n^2\right)}

    3. \sum_{n=-\infty}^{\infty}(-1)^n\exp{\left(-\alpha n^2\right)}

    We have already tried to approximate the summations as an integration of Gaussian. Apparently this is not good enough approximation for our purpose. Any help or suggestions will be appreciated. The first one seems to be key for other two.

    Regards,
    Shantanu
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602

    Re: Summation of e^(-k n^2)

    Quote Originally Posted by mundhadashantanu View Post
    Hi All,

    This is my first thread so please forgive any mistakes regarding forum rules.

    I have been trying to do some calculations for my Physics project. And I get stuck at following summations,

    1. \sum_{n=-\infty}^{\infty}\exp{\left(-\alpha n^2\right)}

    2. \sum_{n=-\infty}^{\infty}n^2\exp{\left(-\alpha n^2\right)}

    3. \sum_{n=-\infty}^{\infty}(-1)^n\exp{\left(-\alpha n^2\right)}

    We have already tried to approximate the summations as an integration of Gaussian. Apparently this is not good enough approximation for our purpose. Any help or suggestions will be appreciated. The first one seems to be key for other two.

    Regards,
    Shantanu
    1. \displaystyle \begin{align*} \sum_{n = -\infty}^{\infty}e^{-\alpha \, n^2} &= \sum_{n = -\infty}^{-1} \left( e^{-\alpha \, n^2} \right) + 1 + \sum_{n = 1}^{\infty} \left( e^{-\alpha \, n^2} \right) \end{align*}

    This entire sum can't possibly converge because if \displaystyle \begin{align*} \alpha > 0  \end{align*}, the terms in the first smaller sum don't go to 0, and if \displaystyle \begin{align*} \alpha < 0 \end{align*}, the terms in the second smaller sum don't go to 0.

    2. Doesn't converge for the same reason.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2012
    From
    India
    Posts
    6

    Re: Summation of e^(-k n^2)

    Hi Prove It,

    Thank you for your reply. I don't quite understand what you mean to say. Did you perhaps miss the fact that it is [TEX]n^2[\TEX] and not n in the exponential? Your first and third terms will give exactly the same value!

    -Shantanu
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602

    Re: Summation of e^(-k n^2)

    Quote Originally Posted by mundhadashantanu View Post
    Hi Prove It,

    Thank you for your reply. I don't quite understand what you mean to say. Did you perhaps miss the fact that it is [TEX]n^2[\TEX] and not n in the exponential? Your first and third terms will give exactly the same value!

    -Shantanu
    Actually you are right, I did miss the \displaystyle \begin{align*} n^2 \end{align*} in the power. Are we assuming that \displaystyle \begin{align*} \alpha > 0 \end{align*}?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2012
    From
    India
    Posts
    6

    Re: Summation of e^(-k n^2)

    Quote Originally Posted by Prove It View Post
    Actually you are right, I did miss the \displaystyle \begin{align*} n^2 \end{align*} in the power. Are we assuming that \displaystyle \begin{align*} \alpha > 0 \end{align*}?
    Yes. \alpha>0 and it is real!

    -Shantanu
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Aug 2011
    Posts
    250
    Thanks
    60

    Re: Summation of e^(-k n^2)

    Hi Shantanu,

    These sums involve the Jacobi theta functions.
    Jacobi Theta Functions -- from Wolfram MathWorld
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Aug 2012
    From
    India
    Posts
    6

    Re: Summation of e^(-k n^2)

    Hi JJacquelin,

    That solves my problem. Thanks a lot for your help!

    -Shantanu
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Summation of summation?
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 3rd 2010, 07:42 PM
  2. summation
    Posted in the Statistics Forum
    Replies: 0
    Last Post: March 6th 2010, 11:27 AM
  3. Summation
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: November 23rd 2009, 04:41 AM
  4. Summation
    Posted in the Algebra Forum
    Replies: 9
    Last Post: August 28th 2009, 07:31 PM
  5. Summation?
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: October 30th 2008, 06:17 PM

Search Tags


/mathhelpforum @mathhelpforum