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Math Help - Sigma Notation from given set of numbers

  1. #1
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    Sigma Notation from given set of numbers

    How would I go about finding sigma notation for any given set of numbers? I really struggle at finding this. Like is there an easy way to write sigma notation just from looking at a set of numbers? Thanks
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    Quote Originally Posted by BCHurricane89 View Post
    How would I go about finding sigma notation for any given set of numbers? Like is there an easy way to write sigma notation just from looking at a set of numbers?
    Please expand on that question.
    Perhaps with an example of exactly what you mean.
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    For example, your given the set of numbers: -2+4-6+8-10

    Write the sum in sigma notation.

    Id like to know how to figure these type of problems out, as I don't quite understand where to start.
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    Quote Originally Posted by BCHurricane89 View Post
    For example, your given the set of numbers: -2+4-6+8-10 Write the sum in sigma notation.
    Examine the pattern.
    Every term is even and the signs alternate.
    \sum\limits_{k = 1}^\infty  {\left( { - 1} \right)^{k } \left( {2k} \right)}
    Using {\left( { - 1} \right)^{k } } is a standard way to alternate signs.

    Quote Originally Posted by BCHurricane89 View Post
    Id like to know how to figure these type of problems out, as I don't quite understand where to start.
    There is no easy answer to that question. Experience and lots of practice is the key.
    Pattern recognition is the most important skill you can have to help you.
    Last edited by Plato; November 12th 2008 at 09:20 AM.
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by BCHurricane89 View Post
    For example, your given the set of numbers: -2+4-6+8-10

    Write the sum in sigma notation.

    Id like to know how to figure these type of problems out, as I don't quite understand where to start.
    There are 5 elements, so our sum will have limits from k=1 to 5.

    Next step is to try to find a pattern.

    We see that the signs alternate, suggesting that there is a (-1)^\alpha term in this sequence. Since the first term is negative, and we're starting from k=1 (which is an odd number), we see that the sequence will include a (-1)^k.

    Now, what is the kth term of the sequence 2,~4,~6,~8,~10?

    We note that this is the sequence of even numbers. and we know that 2(1)=2,~2(2)=4,~2(3)=6,~2(4)=8,~2(5)=10; this suggests that the kth term is 2k.

    We can now see that this sum can be expressed as follows:

    \sum_{k=1}^5 (-1)^k (2k)=2\sum_{k=1}^5 (-1)^k k

    Does this make sense?

    --Chris

    EDIT: Plato beat me. As he mentioned, pattern recognition is very important!!
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    Ok, i kinda see where your going. Whats the difference, because one of you has (-1)^k, and the other has (-1)^(k+1) is there any difference? I have a test today, and this will be on it, and I was looking at my notes, trying to figure out a way to do these, but this helps.


    Oh yeah, also say teh numbers were switched, like 2-4+6-8+10, then it would still be k=1 to 5, then sigma (-1)^k+1 *2k ?
    Last edited by BCHurricane89; November 12th 2008 at 09:24 AM.
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  7. #7
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by BCHurricane89 View Post
    Ok, i kinda see where your going. Whats the difference, because one of you has (-1)^k, and the other has (-1)^(k+1) is there any difference?
    There is a difference if you're starting at k=1.

    If we start at k=1, we see that (-1)^{k+1} will produce the sequence 1,~-1,~1,~-1,~1,~-1,\dots and that (-1)^k will produce the sequence -1,~1,~-1,~1~-1,~1,\dots

    --Chris
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    oh, ok, now I see that
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