Results 1 to 3 of 3

Thread: How to calculate the maximum for a Fourier series?

  1. #1
    Junior Member
    Joined
    Aug 2017
    From
    London
    Posts
    26

    Question How to calculate the maximum for a Fourier series?

    In

    $$\frac{Gt}{j} = \frac{Gj}{R}\Big ( \frac{1}{6}+\frac{2}{\pi^2}
    \sum_{n=1}^{\infty}\frac{(-1)^2}{n^2}
    \exp(\frac{-n^2\pi^2 Rt}{j^2})
    \Big )$$


    How to find the range of $Rt/j^2$ for which the equation is valid?
    Last edited by brianx; Apr 23rd 2018 at 10:52 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    6,363
    Thanks
    2734

    Re: How to calculate the maximum for a Fourier series?

    does $j = \sqrt{-1}$ ?

    I'm assuming it does. Thus we have

    $b_n = \dfrac{e^{n^2\pi^2 R t}}{n^2}$

    is the underlying sequence in the alternating series.

    In order for the series to converge this sequence must have two properties

    i) $b_n$ is ultimately a decreasing sequence

    ii) $\lim \limits_{n \to \infty}~b_n = 0$

    we need the sign of $R t$

    if $R t > 0$ then this sequence tends to infinity

    if $R t \leq 0$ then it tends to 0.

    additionally if $R t \leq 0$ then the sequence is decreasing.

    So it looks like your answer is the formula is valid for $R t \leq 0$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2017
    From
    London
    Posts
    26

    Re: How to calculate the maximum for a Fourier series?

    Quote Originally Posted by romsek View Post
    does $j = \sqrt{-1}$ ?
    no, it is a positive veriable

    Quote Originally Posted by romsek View Post
    does $j = \sqrt{-1}$ ?

    is the underlying sequence in the alternating series.
    yes,

    The entire solution to a partial differential equation is

    $$y(x,t) = \frac{Gt}{j}+\frac{Gj}{D}\Bigg(\frac{3x^2-j^2}{6j^2}-\frac{2}{\pi^2}\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}\exp\Big(\frac{-n^2\pi^2 Rt}{j^2}\Big) \cos\frac{n\pi x}{j}\Bigg)$$

    The above equation is for the case at $x=0$. At a constant $R/j^2$, $y$ increases with $t$. The purpose is to find up to what value of $Rt/j^2$, $y$ remains zero.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to calculate the Fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 10th 2013, 07:25 AM
  2. Is there any way to computationally calculate a fourier series?
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: Jul 15th 2011, 10:31 PM
  3. How do I calculate this Fourier transform?
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: Oct 14th 2010, 07:10 PM
  4. [SOLVED] Fourier series to calculate an infinite series
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Aug 4th 2010, 02:49 PM
  5. Complex Fourier Series & Full Fourier Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Dec 9th 2009, 06:39 AM

/mathhelpforum @mathhelpforum