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Math Help - Appealing to geometric series

  1. #1
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    Appealing to geometric series

    Derive the indicated result by appealing to the geometric series:

    I know that there is something in geometric series that deal with a number being greter than or less than to 1, such as |x|<1 so I know the series must converge, but to what? I know it will converge to a value less than 1...
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    Quote Originally Posted by WartonMorton View Post
    Derive the indicated result by appealing to the geometric series:

    I know that there is something in geometric series that deal with a number being greter than or less than to 1, such as |x|<1 so I know the series must converge, but to what? I know it will converge to a value less than 1...
    Note that the thing you're summing can be written as (-x^2)^k.
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    Quote Originally Posted by mr fantastic View Post
    Note that the thing you're summing can be written as (-x^2)^k.
    What does (-x^2)^k give me? I guess I don't quite understand what form my answer should be in when they say derive the indicated result?
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    Quote Originally Posted by WartonMorton View Post
    What does (-x^2)^k give me? I guess I don't quite understand what form my answer should be in when they say derive the indicated result?
    If |x|<1 then \sum\limits_{k = 0}^\infty  {\left( { - x^2 } \right)^k }  = \frac{1}{{1 + x^2 }}
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