Results 1 to 2 of 2

Thread: What does this sum converge to?

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    36

    What does this sum converge to?

    Hi! I've stumbled upon a convergent sum of the form:

    $\displaystyle S(k) = \sum_{n = 0}^{\infty}{k^{n^2}}$

    This sum converges for every $\displaystyle k$ with $\displaystyle |k| < 1$. I've checked properties of the function numerically (for real values of $\displaystyle k$) and found that for $\displaystyle |k|$ close to zero the function goes approximately as $\displaystyle S(k) = k + 1$. I need to find an expression for the function for $\displaystyle k$ close to 1. I've already found that it goes as $\displaystyle S(k) = C(k)(1 - k)^{\beta}$, where $\displaystyle \beta \approx -\frac{1}{2}$ and $\displaystyle C(k)$ is another function that goes to some finite constant of approximately 0.89 as $\displaystyle k$ goes to 1. I'd be happy if anyone knows an analytic expression of the function (exact or approximate for $\displaystyle k$ close to 1).

    Thanks in advance.

    /Simon
    Last edited by sitho; Feb 23rd 2009 at 04:49 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Feb 2009
    Posts
    36
    Ok, so I've found the approximate behaviour of the function to be $\displaystyle S(k) \approx \frac{1}{2}\sqrt{-\frac{\pi}{\ln(k)}}$ in the limit $\displaystyle k \rightarrow 1^{-}$ by approximating the sum with an integral. I'm not sure however how valid this approximation is in this limit. It fits with numerical estimations though. The constant $\displaystyle \approx 0,89$ is hence $\displaystyle \frac{\sqrt{\pi}}{2} \approx 0,8862...$ I think though that such a fundamental sum should have been studied in detail. Are there any more exact solutions to the problem?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Apr 15th 2011, 05:12 AM
  2. Replies: 4
    Last Post: Jun 26th 2010, 07:04 AM
  3. converge
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Apr 24th 2010, 12:02 PM
  4. converge
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 10th 2009, 11:09 AM
  5. converge or not???
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Nov 22nd 2009, 01:12 AM

Search Tags


/mathhelpforum @mathhelpforum