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Consider the function f(x) = lim_(n -> inf) [ x/(x+1) + x/((x+1)(2x+1)) + x/((x+1)(2x+1)(3x+1)) +... (n terms) ]. What is the range of f(x)?

1. {0, 1}
2. {-1, 0}
3. {-1, 1}
4. [0, 1]

Correct answer: {0, 1}

Solution

When x = 0, every term of the series is 0, so f(0) = 0. When x is not equal to 0 (and x > -1/n for all n, i.e., x > 0 or in valid domain), the first term is x/(x+1) and remaining terms become negligible as n grows since the denominator product diverges; in the limit the partial sums converge to 1 for x > 0 in appropriate analysis. Actually f(x) = 1 for x not equal to 0 and f(0) = 0, giving range {0, 1}.

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