Results 1 to 3 of 3

Math Help - Series of Functions

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    135

    Series of Functions

    This is number 2 of the problem set for real analysis problems:

    Prove that if
    <br />
\sum_{n=1}^{\infty}{{g_n}}<br />
    converges uniformly, then (g_n) converges uniformly to zero.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2008
    Posts
    135

    Ideas

    Intuitively this makes sense to me. If something converges, then that means the terms eventually get small enough so that they are infinitely small. I just can't put it into formal terms through a proof. Any ideas?
    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 ajj86 View Post
    This is number 2 of the problem set for real analysis problems:

    Prove that if
    <br />
\sum_{n=1}^{\infty}{{g_n}}<br />
    converges uniformly, then (g_n) converges uniformly to zero.
    If \sum_{n=1}^{\infty} g_n converges uniformly then it is uniformly Cauchy.
    Let \epsilon > 0 then there is N so that if n,m>N
    Then, \left| \sum_{k=1}^n g_k(x) - \sum_{k=1}^m g_k(x) \right| < \epsilon for x\in S.
    Thus, for example if n=m+1 then, |g_n(x)| < \epsilon.
    Thus, we see that g_n \to 0 uniformly on S.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series of Functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 1st 2011, 06:03 AM
  2. Series of Functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 29th 2010, 04:00 AM
  3. Series of Functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 19th 2008, 07:58 AM
  4. series of functions
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 20th 2008, 12:28 PM
  5. Convergence of a series of functions
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 3rd 2007, 09:23 PM

Search Tags


/mathhelpforum @mathhelpforum