sequence and series
posted by lili .
express the given repeating decimal
0.159159159 as a fraction

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 
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/(1r)
here g = .159
and r = .001
so
S = .159/(1.001) = .159/.999 =159/999