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Algebra Level 4

Suppose SS denotes sum to infinity and SnS_n denotes the sum of nn terms of the series 1+12+14+18+1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots such that SSn<11000S-S_n<\dfrac{1}{1000}, then what will be the least value of nn?

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