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Math Help - Infinite geometric series

  1. #1
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    Infinite geometric series

    Use the sum of an infinite geometric series to prove that o.101010...=10/99. Notice how this decimal expression alternates between 0 and 1 indefinitely, so essentially we do not know whether it terminates on 0 or 1( the fact is, it doesn't terminate). Explain the apparent paradox that this alternating pattern still adds to 10/99.

    What paradox in math means?
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  2. #2
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    Quote Originally Posted by lifeunderwater View Post
    Use the sum of an infinite geometric series to prove that o.101010...=10/99. Notice how this decimal expression alternates between 0 and 1 indefinitely, so essentially we do not know whether it terminates on 0 or 1( the fact is, it doesn't terminate). Explain the apparent paradox that this alternating pattern still adds to 10/99.

    What paradox in math means?
    Strictly speaking "paradox" means something that is both true and not true. Since that is clearly impossible, it is more common to talk about an "apparent" paradox, as done here- something that looks like it can't be true but is.

    Actually, I wouldn't consider this to be at all paradoxical because you can treat it as .10+ 0.0010+ .000010+ ... using blocks of two digits at a time.

    .10+ .0010+ .000010+ ...= .10+ (.10)^2+ (.01)^3+ .... which is a geometric series with "common factor" .10. Do you know the formula for the sum of a geometric series?
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    Quote Originally Posted by HallsofIvy View Post
    .10+ .0010+ .000010+ ...= .10+ (.10)^2+ (.01)^3+ .... which is a geometric series with "common factor" .10. Do you know the formula for the sum of a geometric series?
    Should be \sum\limits_{n = 1}^\infty  {\left( {0.1} \right)^{2n - 1} }  = 0.1 + 0.001 + 0.00001 +  \cdots ?
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