lim _(x → 2) ( Σₙ₌₁⁹ x/(n(n + 1)x² + 2(2n + 1)x + 4) ) is equal to :

Solution

Maths sir · 4 steps

Factor that quadratic denominator and every term splits into a difference, so the whole sum telescopes.

1. Factor the denominator. Check that (nx+2)((n+1)x+2) = n(n+1)x² + 2nx + 2(n+1)x + 4 = n(n+1)x² + 2(2n+1)x + 4. So the denominator is a product of two linear factors.
2. Split each term. The two factors differ by ((n+1)x+2) - (nx+2) = x, which is exactly the numerator. So each term is a difference of two fractions.

   x/(nx+2)((n+1)x+2) = 1/(nx+2) - 1/((n+1)x+2)

3. Telescope the sum. Adding from n = 1 to 9, every middle fraction cancels. Only the first part of n = 1 and the last part of n = 9 survive, giving 1/(x+2) - 1/(10x+2).
4. So. This expression is continuous at x = 2, so put x = 2 directly: 1/4 - 1/22 = (11 - 2)/44 = 9/44. The answer is (a).

The idea to keep. When a term looks like something/AB and B - A equals that something, write it as 1/A - 1/B. The sum then collapses to its first and last pieces.

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