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: :The Geometric Series

  1. #1

    Aug 2011
    Jordan - Zerqa
    901

    :The Geometric Series

    Introduction
    Suppose someone offers you the following deal: You get $1 on the first day, $0.50 the second day, $0.25 the third day, and so on. For a second, you might dream about infinite riches, but adding some of the numbers on your calculator will soon convince you that this is an offer for about $2.00, spread out over quite some time.
    The process of adding infinitely many numbers is at the heart of the mathematical concept of a numerical series.


    Let's see why the deal above amounts to just $2.00. Let s denote the sum of the series just considered:
    Let's multiply both sides by 1/2

    and subtract the second line from the first. All terms on the right side except for the 1 will cancel out! Bingo:

    We have shown that

    One also says that this series converges to 2.

    Let's play the same game for a general q instead of 1/2:
    multiply both sides by q

    then, subtract the second line from the first:


    The series

    is called the geometric series. It is the most important series you will encounter!

    Example:[/h] Find the sum of the series
    First, factor out the 5 from upstairs and a 2 from downstairs:
    .
    The series in the parentheses is the geometric series with , but the first term, the "1" at the beginning is omitted. Thus, the series sums up to

    N.B. There is a slightly slicker way to do this. Do you see how?

    Try it yourself![/h] Find the sum of the series


  2. #2

    Jul 2011
    -
    716

    : :The Geometric Series


  3. #3

    Oct 2011
    20

    : :The Geometric Series


  4. #4

    Jul 2011
    -
    15,678

    : :The Geometric Series




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