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Limit of a Series

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data

    What is the limit of 1/2 Series from (n=1 to n=Infinity) of 1/(n^2 + n).

    3. The attempt at a solution

    This is a simplification of finding the integral of an oscillating function from 0 to 1 that makes triangles of height one between 1/n and 1/(n+1)

    Thus the area of each triangle is 1/2 * (1/n - 1/(n+1)) = 1/(2n^2 + 2n)

    The integral should therefore be equal to the above given limit. From intuition I believe the answer should be 1/2, though I would appreciate any help in determining how this limit is found. Seems like it should be relativley easy to find.
  2. jcsd
  3. Oct 16, 2008 #2


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    Science Advisor

    Well, it's easier in the original form. You want to sum 1/n - 1/(n+1) from n=1 to infinity. Let's write out the first few terms:
    [tex]\left({1\over 1}-{1\over 2}\right)
    +\left({1\over 2}-{1\over 3}\right)
    +\left({1\over 3}-{1\over 4}\right)+\ldots[/tex]
    Notice any way to simplify this?
  4. Oct 20, 2008 #3
    As Avodyne said, it's a telescoping series
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