The value of _n(1)/(n)_j=1^n((2j-1)+8n)/((2j-1)+4n) is equal to :

Question

The value of $\displaystyle\lim_{n	o\infty}\frac{1}{n}\sum_{j=1}^n\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to :

Accepted Answer

$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{\left( {{{2j} \over n} - {1 \over n} + 8} \right)} \over {\left( {{{2j} \over n} - {1 \over n} + 4} \right)}}} $$

$$\int\limits_0^1 {{{2x + 8} \over {2x + 4}}dx = \int\limits_0^1 {dx + \int\limits_0^1 {{4 \over {2x + 4}}} dx} } $$

$$ = 1 + 4{1 \over 2}(\ln |2x + 4|)_0^1$$

$$ = 1 + 2\ln \left( {{3 \over 2}} \right)$$

Solutions · 1