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Telescoping Sum

  1. Mar 9, 2006 #1
    "Let (a_n) be bounded decreasing and (b_n) be bounded increasing sequences. Let x_n =a_n+b_n. Show that [tex]\sum|x_n-x_{n+1}|[/tex] converges."

    This ALMOST is a telescoping sum, but it doesn't work since if I try to use the triangle inequality, the sum I want is on the greater side. Ratio test, root test, etc all fail since there is insufficient information.
  2. jcsd
  3. Mar 9, 2006 #2


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    How did you try to apply the triangle inequality? Try different groupings of the terms.
  4. Mar 10, 2006 #3


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    By the triangle inequality,

    [tex]\sum|x_n-x_{n+1}| = \sum|a_n+b_n-(a_{n+1}+b_{n+1})| \leq \sum|a_n-a_{n+1}|+\sum|b_n-b_{n+1}|[/tex]​

    also, bounded monotonic sequences are convergent.
  5. Mar 10, 2006 #4
    They are indeed convergent, but showing that the terms of the series go to 0 is insufficient to show that it converges. If we remove the absolute value signs, it becomes a telescoping sum and therefore it converges. Knowing that, and that an, bn converge, can we conclude something from it?
  6. Mar 10, 2006 #5


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    Yes, you will have shown the right hand side converges to a finite value. The left hand side has positive terms and is then bounded above, so?
  7. Mar 10, 2006 #6
    How did you conclude that the RHS converges?
  8. Mar 10, 2006 #7
    Nevermind, I figured it out, thanks.
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