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Math Help - Telescoping Series

  1. #1
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    Telescoping Series

    I just need to get a better understanding on what a telescoping series is.

    I think I can recognize it easily enough, but.. please take a look at this:

    "Determine if the following series diverges or converges. If the series converges and it is possible, tell me what the sum of the series is. Identify what kind of series it is or the test you used to validate your answer."

    I don't know how to make all the fancy math symbols, but it's

    (sigma from n=1 to infinity)(5/n - 5/n+1)

    Plugging in the first few values,

    (5/1 - 5/2) + (5/2 - 5/3) + (5/3 - 5/4) ....

    So everything from n+1 to infinity cancels each other out (-5/2 + 5/2, -5/3 + 5/3, etc, etc). We are left with just 5/1, or 5.

    So.. the sum of the series is just 5 + 0 + 0 + 0 ... = 5.. right?

    I would say it converges and the sum is 5. Is this right?

    My notes say the sum is the first number (5/1) minus the limit ... I don't really understand.

    Any help is appreciated.

    Thanks, Jeremy
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  2. #2
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    Quote Originally Posted by JTG2003 View Post
    I just need to get a better understanding on what a telescoping series is.

    I think I can recognize it easily enough, but.. please take a look at this:

    "Determine if the following series diverges or converges. If the series converges and it is possible, tell me what the sum of the series is. Identify what kind of series it is or the test you used to validate your answer."

    I don't know how to make all the fancy math symbols, but it's

    (sigma from n=1 to infinity)(5/n - 5/n+1)

    Plugging in the first few values,

    (5/1 - 5/2) + (5/2 - 5/3) + (5/3 - 5/4) ....

    So everything from n+1 to infinity cancels each other out (-5/2 + 5/2, -5/3 + 5/3, etc, etc). We are left with just 5/1, or 5.

    So.. the sum of the series is just 5 + 0 + 0 + 0 ... = 5.. right?

    I would say it converges and the sum is 5. Is this right?

    My notes say the sum is the first number (5/1) minus the limit ... I don't really understand.

    Any help is appreciated.

    Thanks, Jeremy
    go to the link and scroll down to the middle of the page ...

    Pauls Online Notes : Calculus II - Series - Special Series
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  3. #3
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    Ok, that helps a bit.

    However, why would there be a "Last" term if n goes to infinity?

    In this case, the last term was 5/n+1 which is going to 0 anyways, so 5-0 doesn't really matter.

    Should I just accept this and move on?
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  4. #4
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    Quote Originally Posted by JTG2003 View Post
    However, why would there be a "Last" term if n goes to infinity?
    The answer to that question is simple: Series convergence is all about the sequence of partial sums.
    When are considering a series \sum\limits_{j = 1}^\infty  {\left( {\frac{1}<br />
{j} - \frac{1}{{j + 1}}} \right)} we cannot think of that as a ‘infinite string of numbers’.
    We look at the sequence of partial sums: S_N  = \sum\limits_{j = 1}^N {\left( {\frac{1}<br />
{j} - \frac{1}{{j + 1}}} \right)}  = 1 - \frac{1}{{N + 1}}. (partial sums are finite strings)
    We look at the limit \lim _{N \to \infty } S_n  = \lim _{N \to \infty } \left( {1 - \frac{1}<br />
{{N + 1}}} \right) = 1.

    So series convergence is really about sequence convergence, the sequence of partial sums.
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