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sequence and series

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express the given repeating decimal
0.159159159 as a fraction

  • sequence and series - ,

    n = 0.159159159...

    three numbers 159 repeat
    so multiply by 10^3 = 1000

    1000n =159.159159159 now subtract n
    n = .159159159
    -------------------------------
    999n = 159.000000....
    so
    n = 159/999

  • sequence and series - ,

    or
    159*10^-3 + 159*10^-6 +159*10^-9 ....
    each term is 10^-3 times previous
    sum of infinite geometric series with first term g and ratio r
    S = g/(1-r)
    here g = .159
    and r = .001
    so
    S = .159/(1-.001) = .159/.999 =159/999

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