Question

The value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n }$ $\sum_{r = 1}^{n}\left( \frac{r}{n + r} \right)$is

• A. ln 2
• B. 1 + ln 2
• C. 1 – ln 2
• D. 0

Answer 1

Answer: (C)

1 – ln 2

Answer 2

$\lim _ { n \rightarrow \infty }$ $\frac{1}{n}\sum_{r = 1}^{n}\frac{r}{n + r}$ = $\lim _ { n \rightarrow \infty }$ $\frac{1}{n}\sum_{r = 1}^{n}\left( \frac{\frac{r}{n}}{1 + \frac{r}{n}} \right)$

= $\int_{0}^{1}\frac{xdx}{1 + x}$ = $\int_{0}^{1}{dx}$ – $\int_{0}^{1}\frac{dx}{1 + x}$ = 1 – ln 2