Results 1 to 6 of 6

Math Help - Series Uniform Continuity proof

  1. #1
    Junior Member
    Joined
    Nov 2010
    Posts
    40

    Series Uniform Continuity proof

    Let f(x)= sum [(3^(n) + cos(n))/n!]X^n

    1. Prove that for every x in [-10,10] the sum converges
    2. Show that for every E >0 there's an N independant of x in [-10,10] such that
    |f(x) - sum [(3^(n) + cos(n))/n!]X^n | < E
    3. Use 2 together with the fact that polynomials are contiuous everywhere to show that f is continuous in [-10,10]

    ratio test for part 1, not sure how to fully show it
    2 I'm not sure what to do here

    the sum in part two is from 0 to N
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by gutnedawg View Post
    Let f(x)= sum [(3^(n) + cos(n))/n!]X^n
    ratio test for part 1, not sure how to fully show it
    If the series is \sum_{n=0}^{+\infty}u_n(x) then, express:

    \left |{\dfrac{u_{n+1}(x)}{u_n(x)}}\right |=\left | \dfrac{3+\cos (n+1)/3^{n+1}}{1+\cos n/3^{n+1}}\cdot{\dfrac{1}{n+1}} \right|\left |{x}\right |\rightarrow{0}\;(n\rightarrow{+\infty})

    This series converges on \mathbb{R}, as a consequence on [-10,10]

    2 I'm not sure what to do here
    the sum in part two is from 0 to N
    Find a bound |u_n(x)|\leq a_n on [-10,10], prove that \sum_{n=0}^{+\infty}a_n converges and use Cauchy's criterion of convergence.

    Regards.

    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    how would this work in part 2? I was under the impression that it would be just like a limit proof of sorts with the difference in the abs being sum N to infinity of * and then solve for N then using n>N show the proof.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by gutnedawg View Post
    how would this work in part 2? I was under the impression that it would be just like a limit proof of sorts with the difference in the abs being sum N to infinity of * and then solve for N then using n>N show the proof.
    On [-10,10] we have:

    \left |{u_n(x)}\right |\leq \ldots=\dfrac{3^n+1}{n!}10^n=\alpha_n

    and the series \sum_{n=0}^{+\infty}\alpha_n <br />
is convergent. So, \sum_{n=0}^{+\infty}u_n(x) <br />
uniformly converges to f(x) on [-10,10] as a consequence of the Weierstrass criterion (which is deduced from Cauchy criterion).

    Regards.

    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    I am not allowed to use anything to do with uniform convergence, this would not be difficult if I was allowed to use these theorems =[
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Post

    No problem, as \sum_{n=o}^{\infty}\alpha_n is convergent,

    \forall \epsilon >0\;\exists n_0\in \mathbb{N}: n\geq m\geq n_o\Rightarrow\alpha_m+\alpha_{m+1}+\ldots+\alpha_  n<\epsilon

    Then

     n\geq m\geq n_0\Rightarrow|u_m(x)|+|u_{m+1}(x)|+\ldots+|u_n(x)  |<\epsilon for all x\in [-10,10]

    A fortiori,

     n\geq m\geq n_0\Rightarrow|u_m(x)+u_{m+1}(x)+\ldots+u_n(x)|<\e  psilon for all x\in [-10,10]

    Could you continue?.

    Regards.

    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniform Continuity Proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 1st 2010, 07:46 PM
  2. uniform continuity proof
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 27th 2009, 05:36 PM
  3. Uniform Continuity Proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 18th 2007, 05:23 PM
  4. Uniform Continuity Proof 2
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 16th 2007, 05:28 PM
  5. Uniform Continuity Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 15th 2007, 03:42 AM

Search Tags


/mathhelpforum @mathhelpforum