Results 1 to 3 of 3

Math Help - integral of a sequence of functions

  1. #1
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1

    integral of a sequence of functions

    Prove that if on an interval [a,b] the functions g_{n}(x) are continuous and converge uniformly to g(x), then on [a,b]

    \int_a^x g_{n}(t) dt converges uniformly to \int_a^x g(t) dt.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Abu-Khalil's Avatar
    Joined
    Oct 2008
    From
    Santiago
    Posts
    148
    g(x) is continuous and therefore integrable over [a,b] 'cause g_n\to g uniformly and each g_n is continuous there.

    Let f_n(x)=\int_a^xg_n(t)dt,f(x)=\int_a^xg(t)dt. So we wanna prove that f_n\to f uniformly.

    Let \epsilon>0, then \exists N:\forall n\geq N,\left|g_n(x)-g(x)\right|<\epsilon,\forall x\in[a,b].
    Notice that \forall x\in[a,b] we have \left|f(x)-f_n(x)\right|=\left|\int_a^xg_n(t)dt-\int_a^xg(t)dt\right|\leq \int_a^x\left|g_n(t)-g(t)\right|dt\leq \int_a^x\epsilon dt\leq \epsilon(b-a).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Anonymous1 View Post
    Prove that if on an interval [a,b] the functions g_{n}(x) are continuous and converge uniformly to g(x), then on [a,b]

    \int_a^x g_{n}(t) dt converges uniformly to \int_a^x g(t) dt.



    \forall\,\epsilon>0\,\,\exists\,N_{\epsilon}\!\!\i  n\mathbb{N}\,\,s.t.\,\,\left|g_n(x)-g(x)\right|<\frac{\epsilon}{b-a}\,\,\forall\,n>N_{\epsilon}\mbox{ and }\forall\,x\in[a,b] , so :

    \left|\int\limits_a^x g_{n}(t) dt-\int\limits_a^x g(t) dt\right|\leq \int\limits_a^b \left|g_n(t)-g(t)\right|dt<\int\limits_a^b\frac{\epsilon}{b-a} dt=\epsilon\,,\,\,\forall\,n>N_{\epsilon} \mbox{ and }\,\forall\,x\in[a,b].

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Example of sequence of functions
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: May 1st 2011, 01:11 PM
  2. Sequence of functions
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 3rd 2011, 06:37 PM
  3. example of sequence of functions
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 24th 2011, 01:52 PM
  4. Sequence of functions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 7th 2010, 07:37 AM
  5. sequence of functions
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 14th 2008, 05:16 PM

Search Tags


/mathhelpforum @mathhelpforum