1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Uniform convergence integration

  1. Nov 14, 2012 #1
    f is a continuous function on [0,infinity) such that 0<=f(x)<=Cx^(-1-p) where C,p >0 f_k(x) = kf(kx)

    I want to show that lim k->infinity ∫from 0 to 1 of f_k(x) dx exists

    so my idea is if I have that f_k(x) converges to f(x)=0 uniformly which I was able to show and that f_k(x) are all integrable, then the limit can be moved inside and so this thing will exist.

    to show f_k(x) is integrable from 0 to 1, I replaced it with Cx^(-1-p) and ran into a problem, this is only integrable from 0 to 1 when the exponent (1+p) on the bottom now needs to be <1 which requires p to be less than 0 but p is given to be > 0. is my understanding of integrability of improper integrals correct or is it something else I'm doing wrong. Thanks a lot.
  2. jcsd
  3. Nov 14, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    You don't have to worry much about what happens on [0,1] particularly near 0. You are given that f is continuous on [0,1]. It integrable on [0,1]. f is much better behaved there than your bounding function is.
  4. Nov 18, 2012 #3
    thanks that's very helpful!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook