lim _(n → ∞) Σᵣ₌₁ⁿ r/(2r² - 7rn + 6n²) is equal to :

Solution

Maths sir · 4 steps

Divide top and bottom by n² so everything is in r/n, and the sum turns into ∫₀¹.

1. Make it r/n. Divide the numerator and the denominator by n². Each term becomes 1/n · (r/n)/(2(r/n)² - 7(r/n) + 6). Now replace r/n by x and 1/n by dx. Since r runs from 1 to n, x runs from 0 to 1.

   L = ∫₀¹ x/(2x² - 7x + 6) dx = ∫₀¹ x/(2x-3)(x-2) dx

2. Partial fractions. Write x/(2x-3)(x-2) = A/(2x-3) + B/(x-2), so x = A(x-2) + B(2x-3). Putting x = 2 gives B = 2. Putting x = 3/2 gives 3/2 = -A/2, so A = -3.
3. Integrate. The antiderivative is -3/2 ln|2x-3| + 2 ln|x-2|. At x = 1 it is 0. At x = 0 it is -3/2 ln 3 + 2 ln 2.
4. So. L = 3/2 ln 3 - 2 ln 2 = ln 3√3/4, which is option (b).

The idea to keep. When the bottom has degree 2 in r and n and the top has degree 1, divide by n². One spare 1/n becomes dx, and r/n becomes x from 0 to 1.

Answer from the NTA final answer key. Solution by PrepWiser.