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Evaluate the limit as n approaches infinity of the expression [1*n + 2*(n-1) + 3*(n-2) +... + n*1] divided by [1² + 2² + 3² +... + n²].

1. 1
2. 1/3
3. 1/4
4. 1/2

Correct answer: 1/2

Solution

The numerator equals n(n+1)(n+2)/6 and the denominator equals n(n+1)(2n+1)/6, so the ratio is (n+2)/(2n+1), which tends to 1/2 as n -> infinity.

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