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


    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!
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