1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence of integral

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

    Problem:
    -------
    Define

    [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,

    W.
     
  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,

    W.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Convergence of integral
  1. Convergent integrals (Replies: 1)

Loading...