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Help with serie of function

  1. Mar 31, 2005 #1

    quasar987

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    Hi, here the question I have to answer.

    Consider the serie of function

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

    a) For which values of x does the serie converges pointwise?

    b) Over which intervals of the form [a,b] does the serie converges uniformally? Are there cases where we can have a uniform convergence for a=-infty or b = +infty ?


    I answered to a) using the integral convergence criterion for numerical series, but I'm not sure we're allowed to use it since we haven't seen it in class. Is there another way to answer a) ? I first tried to find the form of the nth term of the sequence of partial sums, but I gave up quickly, being unable to see the patern in the sequences {0, 5, 28, 90, ...}, {0, 0, 49, 546,...}, {0, 0, 0, 820, ...}, etc. representing the coefficient of x, x², x^3, etc. respectively.


    For b), I found that for x in (0,infty), the serie converges uniformly. I did so by comparing (through the Weierstrass criterion) the serie with the serie of 1/n², which converges. I don't see how to go about testing the other intervals of pointwise convergence for uniform convergence.

    Thanks for your time and help.
     
  2. jcsd
  3. Apr 2, 2005 #2

    quasar987

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    If I try to find the form of the nth term of the partial sum, I get

    [tex]S_1(x) = \frac{1}{1+x}[/tex]

    [tex]S_2(x) = \frac{2+5x}{(1+x)(1+4x)}[/tex]

    [tex]S_3(x) = \frac{3+28x+49x^2}{(1+x)(1+4x)(1+9x)}[/tex]

    [tex]S_4(x) = \frac{4+90x+546x^2+820x^3}{(1+x)(1+4x)(1+9x)(1+16x)}[/tex]

    ...

    [tex]S_n(x) = \frac{n+}{(1+x)(1+4x)(1+9x)(1+16x)...(1+n^2x)}[/tex]

    Does anyone sees the patern here? :confused:
     
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