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Let alpha = lim(n->inf) (1² + 2² + 3² +... + n²) / n³ and beta = lim(n->inf) [(1³ - 1²) + (2³ - 2²) +... + (n³ - n²)] / n⁴. Which of the following relations between alpha and beta is correct?

1. alpha = beta
2. 4*alpha = 3*beta
3. 3*alpha = 4*beta
4. None of these

Correct answer: 3*alpha = 4*beta

Solution

Evaluating the limits gives alpha = 1/3 and beta = 1/4; then 3*alpha = 1 and 4*beta = 1, so 3*alpha = 4*beta.

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