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20 June, 01:02

Find the sum of the infinite geometric series: 9 - 6 + 4 - ...

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  1. 20 June, 01:28
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    The defining characteristic of all geometric sequences is a common ratio which is a constant when dividing any term by the term preceding it.

    In this case the common ratio is: - 6/9=4/-6=r=-2/3

    An infinite series will have a sum when r^2<1, so in this case the sum will converge to an actual value because (-2/3) ^ (+oo) approaches zero.

    The sum of any geometric sequence is:

    s (n) = a (1-r^n) / (1-r), since we have a common ratio of - 2/3 and we want to calculate an infinite series, ie, n approaches infinity, the sum becomes simply:

    s (n) = a / (1-r) (because (1-r^+oo) approaches 1 as n approaches + oo)

    So our infinite sum is:

    s (+oo) = 9 / (1--2/3)

    s (+oo) = 9 / (1+2/3)

    s (+oo) = 9 / (5/3)

    s (+oo) = 27/5

    s (+oo) = 54/10

    s (+oo) = 5.4
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