Results 1 to 3 of 3

Math Help - Prove this series of functions convergent uniformly

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    Prove this series of functions convergent uniformly

     (u_{n}(x) : n=0,1,...) be a sequence of real-valued functions on subset E of R.

    suppose for all  x\in E, |u_{n}(x)|\leq M_{n}

    where  \sum_{n=0}^{\infty}{M_{n}} converges.

    prove that:  \sum_{0}^{\infty}{u_{n}(x)} converges uniformly on E.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by szpengchao View Post
     (u_{n}(x) : n=0,1,...) be a sequence of real-valued functions on subset E of R.

    suppose for all  x\in E, |u_{n}(x)|\leq M_{n}

    where  \sum_{n=0}^{\infty}{M_{n}} converges.

    prove that:  \sum_{0}^{\infty}{u_{n}(x)} converges uniformly on E.
    A sequence of real-valued functions on E is uniformly convergent if and only if it is uniformly Cauchy.
    That is a fact used in the link Opalg gave above.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. uniformly convergent
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 31st 2011, 04:58 AM
  2. Replies: 3
    Last Post: May 21st 2011, 01:19 PM
  3. Prove that a series is convergent
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 7th 2011, 02:37 AM
  4. Uniformly convergent sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 16th 2009, 08:50 PM
  5. Replies: 1
    Last Post: October 18th 2008, 03:50 PM

Search Tags


/mathhelpforum @mathhelpforum