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Convergence of infinite sequences

  1. Oct 7, 2014 #1
    1. The problem statement, all variables and given/known data
    Let V consist of all infinite sequences {xn} of real numbers for which the series summation xn2 converges. If x = {xn} and y = {yn} are two elements of V, define (x,y) = summation (n=1 to infinity) xnyn.
    Prove that this series converges absolutely.

    2. Relevant equations
    The question includes a Hint: Use the Cauchy-Schwarz inequality to estimate the sum, summation (n=1 to M) lxnynl.

    3. The attempt at a solution
    Using the definition of convergence and ineqaulities I have shown that since xn converges, we have the inequality, summation(xn) < summation (xn2) < infinity. Therefore, {xn} converges but I do not know how to use the hint to extend this to the inner product.
     
  2. jcsd
  3. Oct 7, 2014 #2

    pasmith

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    So what is the Cauchy-Schwartz inequality? Your aim is probably to show that [itex]\sum |x_n y_n| \leq \left(\sum x_n^2\right)^{1/2}\left(\sum y_n^2\right)^{1/2}[/itex]. Why is that enough for you to conclude that [itex]\sum |x_ny_n|[/itex] converges?

    This is false. If [itex]0 < x_n < 1[/itex] then [itex]x_n > x_n^2[/itex]; and in particular you should be aware that [itex]\sum_{n=1}^\infty 1/n[/itex] diverges, whereas [itex]\sum_{n=1}^\infty 1/n^2 = \pi^2/6[/itex]. Even if it were true it wouldn't assist you.
     
    Last edited: Oct 8, 2014
  4. Oct 7, 2014 #3

    RUber

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    You could also note that for any choice of x and y in V, ## |x_n| |y_n| \leq \max( x_n^2, y_n^2 ), \, \forall n \in \mathbb{N} ##. This completely disregards your hint, but in my mind is pretty straightforward.
     
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