Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Series of functions help.

  1. Feb 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the series,


    defines a continuous function f on the domain of convergence. What is this domain? In addition, write a series representation of ( f ' ) and determine the domain of convergence of this series to ( f ' ).

    2. Relevant equations

    3. The attempt at a solution

    I need abit of help with this problem. If somebody could point me in the direction I would be very happy.

    It looks to me that the series in question might be smaller than the series 1/ (k^2) and therefore converges on a domain of all real numbers.

    I had a question about the wording, "the series defines a continuous function f on the domain of convergence". Does this mean that I am looking for the function that this series uniformly or pointwise converges to?
  2. jcsd
  3. Feb 24, 2009 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    In a manner of speaking, but you can consider it more directly. If, given a value of x, the sum converges, then we can define f(x) = the value of the summation for x, and this is the function they're referring to.
  4. Feb 25, 2009 #3
    I have that for,

    [tex]x=0 , \sum_{n=1}^{\infty}\frac{1}{n^2}[/tex]

    [tex]x=1 , \sum_{n=1}^{\infty}\frac{1}{1+n^2}[/tex]

    [tex]x=2 , \sum_{n=1}^{\infty}\frac{1}{4+n^2}[/tex]

    I know that these converge to 0. So no matter what are x values, this series converges. I would say that the domain of convergence is all of R. Am I right with that?

    So did the series define the function f(x)=0. I am having trouble understanding what this means.
  5. Feb 26, 2009 #4
    Ok, so this series converges to some function f and I must find what the interval of x is in which this happens.

    Second, if we take the derivative of f, say ( f ' ), the same series also converges to ( f ' ) but on a different (or same) interval ?
  6. Feb 26, 2009 #5


    User Avatar
    Homework Helper

    why not have a look at the term by term differentiation of f? i think it asks for this is the question
    Last edited: Feb 26, 2009
  7. Feb 26, 2009 #6


    User Avatar
    Science Advisor

    Surelly you know that showing that a function converges for 3 values of x does NOT show that it converges for ALL x. Also, it is clear that those DO NOT "convert to 0". The very first one starts 1+ 1/4+ 1/9 and has only positive terms.
  8. Feb 26, 2009 #7
    I have a theorem that says that the radius of convergence of a power series is the same as the radius of convergence of the term by term differentiation of the power series.

    I didn't know if this was a power series or not. I cannot see how to separate the a_n and the x^n if it is, and if it isn't does the result apply also to series like the one I have brought up?

    I am confused about this though. I though that I could show that the summation for 1/k^2 converges by the ratio test by proving that for n large enough the terms of the ratio are all less than 1. Then for any other x in R, x^2 is positive and would be adding more to the denominator, thus making it even smaller and thus would converge also.

    What am I confusing here? I understand that we are adding positive terms now though, and the series all converge to something that is not necessarily 0.
  9. Feb 26, 2009 #8
    I understand the problem better now. I don't think I need to find the equation of the function that it converges to. (It would look like a bell curve though, I'm guessing)

    I used the ratio test (with the x's in there) and determined that the series converges for all x in R. (R = infinity). I then did a term by term differentiation and performed the ratio test on that to get the same radius of convergence (all x in R).

    I've learned a couple things here about the function. Is there anything else I am missing? (Assuming that doing the 2 ratio tests were valid?)
  10. Feb 26, 2009 #9


    User Avatar
    Homework Helper

    not sure what your lecturer wants.... but you have shown both f & f' converge on a radius of covergence, and determined this radius

    Reading the question I think you might still need to show continuity on the domain of convergence... maybe a quick epsilon delta?
  11. Feb 26, 2009 #10
    I was wondering if I could do that. Do you mean something like this?

    [tex]| \sum_{n=1}^{\infty}\frac{1}{x^2+n^2}- \sum_{n=1}^{\infty}\frac{1}{a^2+n^2}|[/tex]

    I tried this to no avail after I combined? the series into one. I found one |x-a| in there but it seemed like nothing else.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook