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!

Product of two sequences of functions [uniform convergence]

  1. Oct 20, 2011 #1
    1. The problem statement, all variables and given/known data
    This is a homework question for a introductory course in analysis. given that
    a) the partial sums of [itex]f_n[/itex] are uniformly bounded,

    b) [itex] g_1 \geq g_2 \geq ... \geq 0, [/itex]

    c) [itex] g_n \rightarrow 0 [/itex] uniformly,

    prove that [itex]\sum_{n=1}^{\infty} f_n g_n [/itex] converges uniformly (the whole adventure takes place on some interval E in R).

    2. Relevant equations
    Suppose [itex]x[/itex] and [itex]y[/itex] are two sequences. Then,

    [itex] \sum_{j=m+1}^{n} x_jy_j = s_ny_{n+1} - s_my_{m+1} + \sum_{j=m+1}^{n} s_j(y_j - y_{j+1}). [/itex]

    This is called partial summation, and is given as a hint with the exercise.

    3. The attempt at a solution
    Inspired by the Cauchy-criterion for uniform convergence of series of functions, I did the following.

    [itex] | \sum_{j=m+1}^{n} f_n g_n | = | (\sum_{i=1}^{n}) f_i g_{n+1} - (\sum_{i=1}^{m} f_i) g_{m+1} + \sum_{j=m+1}^{n} (\sum_{i=1}^{j} f_i) (g_j - g_{j_1} ) | [/itex]

    [itex] \leq |g_{n+1} \sum_{i=1}^{n} f_i| + |g_{m+1} \sum_1^m f_i | + | \sum_{j=m+1}^{n} (\sum_{i=1}^{j} f_i) (g_j - g_{j_1} ) |[/itex]
    (the last step owing to the subadditivity of the modulus).
    The first two terms can be made small since the partial sums of [itex]f[/itex] are bounded and g goes to zero, leaving the third term. I'm having trouble doing anything interesting with that though. Am I on the right track?
     
    Last edited: Oct 20, 2011
  2. jcsd
  3. Oct 20, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You can bound [itex]\sum{f_j}[/itex] by L. This leaves you with

    [tex]L\sum{g_j-g_{j+1}}[/tex]

    But look at this sum carefully. Isn't that a telescoping sum??
     
  4. Oct 21, 2011 #3
    thanks a lot! I tried to do the same thing but couldn't get [itex] g_i - g_{i+1} [/itex] to converge. I feel somewhat silly now!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Product of two sequences of functions [uniform convergence]
Loading...