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Evaluate the limit: lim (n -> inf) [ 1/(sqrt(n)*sqrt(n+1)) + 1/(sqrt(n)*sqrt(n+2)) +... + 1/(sqrt(n)*sqrt(5n)) ]

1. 2*(sqrt(2) - 1)
2. 4
3. 2*sqrt(2) - 1
4. 2*(sqrt(5) - 1)

Correct answer: 2*(sqrt(5) - 1)

Solution

The sum equals (1/n)*sumₖ₌₁⁴ⁿ 1/sqrt(1+k/n), which as n->inf is the Riemann integral integral₀⁴ (1+x)^(-1/2) dx = [2*sqrt(1+x)]₀⁴ = 2*sqrt(5) - 2 = 2*(sqrt(5)-1).

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