A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index . The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series.
For the simplest case of the ratio equal to a constant , the terms are of the form . Letting , the geometric sequence with constant is given by
(1)
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is given by
(2)
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Multiplying both sides by gives
(3)
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and subtracting (3) from (2) then gives
(4)
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(5)
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so
(6)
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For , the sum converges as ,in which case
(7)
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Similarly, if the sums are taken starting at instead of ,
(8)
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(9)
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the latter of which is valid for .