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Math Help - summation problem

  1. #1
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    summation problem

    I'm trying to figure out this problem and have no clue how to do it. The problem reads:

    Use the identity 1/(k^2-1) = (1/2)(1/(k-1) - 1/(k+1)) to evaluate sum k=2 to n 1/(k^2-1)
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  2. #2
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    the problem is almost solved, now, we have \frac{1}{k-1}-\frac{1}{k+1}=\left( \frac{1}{k-1}-\frac{1}{k} \right)+\left( \frac{1}{k}-\frac{1}{k+1} \right). Then, if you consider <br />
\sum\limits_{k=2}^{n}{\left( \frac{1}{k-1}-\frac{1}{k} \right)}, does it telescope? what 'bout the other one?
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  3. #3
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    what do you mean "does it telescope"? sorry, im new at this stuff.
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  4. #4
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    Quote Originally Posted by geoffl View Post
    what do you mean "does it telescope"? sorry, im new at this stuff.
    Write out the first few terms of the series posted by Krizalid. What do you notice ....?
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by geoffl View Post
    what do you mean "does it telescope"? sorry, im new at this stuff.
    What he means is that a telescoping sum is a sum that can be simplified by the cancelleling of terms through the addition that it implies.

    e.g.

    \sum_{i=1}^n(a_n-a_{n-1})

    If you look carefully at this sum you can see that no matter how large n is that there will only be two terms left standing after what I call the addition wars. check it out

    \sum_{i=1}^n(a_n-a_{n-1})

    =(a_n-a_{n-1})+(a_{n-1}-a_{n-2})+(a_{n-2}-a_{n-3})+...+(a_{3}-a_{2})+(a_{2}-a_{1})

    So after an uncountable number of n terms, who comes out of this war alive?

    \sum_{i=1}^n(a_n-a_{n-1})=a_{n}-a_{1}

    So now look at your problem with a trained eye. You have the ability to predict who's side you shoulb be on!
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  6. #6
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    ahhh, ok. that makes sense.

    thanks a lot!
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