[ _x 2 ( _n = 1^9 x n(n + 1)x^2 + 2(2n + 1)x + 4 )] is equal to :

Question

$$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$$ is equal to :

Accepted Answer

$$S = \mathop {\lim }\limits_{x \to 2} \sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} $$

$$S = \sum\limits_{n = 1}^9 {{2 \over {4({n^2} + 3n + 2)}}}  = {1 \over 2}\sum\limits_{n = 1}^9 {\left( {{1 \over {n + 1}} - {1 \over {n + 2}}} \right)} $$

$$S = {1 \over 2}\left( {{1 \over 2} - {1 \over {11}}} \right) = {9 \over {44}}$$

Solutions · 1