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: Another uniform convergence question

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known data

    h:[0,1] -> R is continuous
    Prove that t(x) = [tex]\sum[/tex][tex]^{infinity}_{n=0}[/tex] xnh(xn) is uniformly convergent on [0,s] where 0<s<1

    2. Relevant equations



    3. The attempt at a solution

    I have the definition of h being continuous but after this I am pretty clueless about how to tackle this problem. I could use the Weierstrass M-test. I know the series xn converges uniformly on this interval as xn < sn but I don't know how to use the fact that h is continuous to find a sequence of real numbers that xnh(xn) is always less than.
     
  2. jcsd
  3. Feb 14, 2010 #2
    Use the extreme value theorem to bound h(x^n), and then find small bounds for x^n h(x^n).
     
  4. Feb 14, 2010 #3
    h is continous (it is a continous function of a polynomial which is continuous) on a compact set therefore h is bounded for x in the interval [0,s]. the sequence of h's for all n is uniformly bounded by a single M. as you said the s^n is a geometric series so it converges. hence uniform convergence.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook