MCQ
$\mathop {\lim }\limits_{n \to \infty } \,\frac{{\sum\limits_{r = 0}^n {{{\tan }^{ - 1}}\left( {1 + r + {r^2}} \right)} }}{n}$ is equal to
- A$1$
- B$2$
- C$\frac {\pi}{4}$
- ✓$\frac {\pi}{2}$
$ = \sum\limits_{r = 0}^n {\frac{\pi }{2} - {{\tan }^{ - 1}}} \frac{1}{{1 + r + {r^2}}}$
$ = \frac{{n\pi }}{2} - \sum\limits_{r = 0}^n {{{\tan }^{ - 1}}} \frac{{\left( {r + 1} \right) - r}}{{1 + r\left( {r + 1} \right)}}$
$ = \frac{{n\pi }}{2} - \sum\limits_{r = 0}^n {\left( {{{\tan }^{ - 1}}\left( {r + 1} \right) - {{\tan }^{ - 1}}r} \right)} $
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Which of the following statements is/are $TRUE$ ?
$(A)$ No $a$ satisfies the above equation
$(B)$ An integer $a$ satisfies the above equation
$(C)$ An irrational number $a$ satisfies the above equation
$(D)$ More than one $a$ satisfy the above equation