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Convergence of integral

  1. Aug 9, 2005 #1
    Hello, it's me again ;)


    [tex] f_{n}(x)=n^{\alpha}, |x|\leq 1/n, f_{n}=0[/tex] elsewhere

    Give all [tex]\alpha \in \Re [/tex] for which

    [tex] \lim_{n \to \infty} \int_{\Re}f_{n}(x)dx=+\infty [/tex]

    Can i change this last integral to:

    [tex] \lim_{x \to 0} \int_{0}^{\infty} x^{-\alpha}dx=+\infty [/tex]

    But i think the integration limits aren't correct, and therefore [tex] alpha [/tex] is wrong too.

    Any help appreciated,

    kind regards,

  2. jcsd
  3. Aug 10, 2005 #2
    Let's see, your function f_n is a constant n^a on the interval [-1/n, 1/n], and zero elsewhere. So the integral is just

    [tex] \int_{\Re}f_{n}(x)dx= n^a \cdot \frac{2}{n}= 2 n^{a-1} [/tex]

    as you can easily see from the graph of f_n.

    You can take it from there...
  4. Aug 10, 2005 #3
    Off course... now that I see it, it's all very simple. Guess sometimes my mind gets confused after too much maths ;).
    Thank you,

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