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QUESTION: Interval of Convergens for a series

  1. May 10, 2006 #1

    I have this series here

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

    I need to show that the Radius of convergens [tex]R = \infty [/tex] and the interval of convergens therefore is [tex](- \infty, \infty) [/tex]

    My question is to do this don't I use the ratio-test?

    Sincerely Yours

    Last edited: May 10, 2006
  2. jcsd
  3. May 10, 2006 #2
    a_n is the nth term, so:

    [tex]\left| \frac{a_{n+1}}{a_n} \right | = \left| \frac{1}{x^2+(n+1)^2} \cdot \frac{x^2+n^2}{1} \right |[/tex]

    However, this goes to 1 as n goes to inifinity so this really doesn't help you. What you probably want to do is use comparison test with 1/n^2 to show that it converges for any x.
    Last edited: May 10, 2006
  4. May 10, 2006 #3
    Hi and thanks for Your answer,

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

    Then by the comparison test:

    [tex]\frac{1}{x^2 + n^2} < \frac{1}{n^2}[/tex] ??



  5. May 10, 2006 #4
    Thats right, and since [tex]\sum_{n=1} ^{\infty} \frac{1}{n^2}[/tex] converges (by p-series), [tex]\sum_{n=1} ^{\infty} \frac{1}{x^2+n^2}[/tex] converges since every term is smaller.
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