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Math Help - Can someone explain Geometric series?

  1. #1
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    Can someone explain Geometric series?

    This may be the wrong place to post this, but I'm going to ask here anyway...

    For ages I've seen people using the sum ( \sum) symbol with like n-1 on top and i=2 beneath it (or something along those lines), and I understand this is a geometric series, and I feel I almost understand it, but I thought if someone could try and explain it I wouldn't get confused ^_^

    I'm probably going to ask my maths teacher anyway, but as it's a Sunday I thought I'd ask on here...

    Any help much appreciated I missed the lesson on this in school so I don't fully understand


    Oh and move this to a more relevant place if this is the wrong place to post
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  2. #2
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    A geometric series is a sequence where the there is a constant ratio between each term.

    eg, the geometric sequence where the multiplier is 1/2 and the first term is 3 would look like:

    3, \frac{3}{2} , \frac{3}{4}, \frac{3}{8} , \frac{3}{16},  \frac{3}{16}, \frac{3}{32}, \frac{3}{64}, \frac{3}{128}...




    Suppose we wanted to find the sum of all terms in the series. We would do:

    3 + \frac{3}{2} + \frac{3}{4}+ \frac{3}{8} + \frac{3}{16},   \frac{3}{16}+ \frac{3}{32}+ \frac{3}{64}+ \frac{3}{128}...

    unfortunately, you would never stop writing the sum down!

    Instead, we notice that the series can be written:
    3 + \frac{3}{2} + \frac{3}{2^2}+ \frac{3}{2^3} + \frac{3}{2^4}...

    =\frac{3}{2^0} + \frac{3}{2^1} + \frac{3}{2^2}+ \frac{3}{2^3} +  \frac{3}{2^4}...

    =3\left(\frac{1}{2^0} + \frac{1}{2^1} + \frac{1}{2^2}+ \frac{1}{2^3} +   \frac{1}{2^4}... \right)

    Which can be expressed as a single summation

    3\sum_{i=0}^{\infty}\frac{1}{2^i}

    is that what you were asking? or did you want an explanation of why 3\sum_{i=0}^{\infty}\frac{1}{2^i} = 3\left(\frac{1}{2^0} + \frac{1}{2^1} + \frac{1}{2^2}+ \frac{1}{2^3} +   \frac{1}{2^4}... \right). in that case you might want to look at http://en.wikipedia.org/wiki/Summation
    Last edited by SpringFan25; June 6th 2010 at 02:57 PM.
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  3. #3
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    Thanks

    That really helped, and yeah I was wondering why http://www.mathhelpforum.com/math-help/JSREPL85139:; ... But that example is simple and easy to understand so now I can like.... you know ^_^ anyway THANKS.









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