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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