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Thread: 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 $\displaystyle (-x^2)^k$.
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    Quote Originally Posted by mr fantastic View Post
    Note that the thing you're summing can be written as $\displaystyle (-x^2)^k$.
    What does $\displaystyle (-x^2)^k$ 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 $\displaystyle (-x^2)^k$ give me? I guess I don't quite understand what form my answer should be in when they say derive the indicated result?
    If $\displaystyle |x|<1$ then $\displaystyle \sum\limits_{k = 0}^\infty {\left( { - x^2 } \right)^k } = \frac{1}{{1 + x^2 }}$
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