_ r = 1 ^ ^ - 1 ( 3 r^2 - r + 9 ) is- - Vidyadip

MCQ

$\sum\limits_{r = 1}^\infty {{{\tan }^{ - 1}}\left( {\frac{3}{{{r^2} - r + 9}}} \right)} $ is-

A
$\frac{\pi }{3}$
B
$\frac{\pi }{6}$
✓
$\frac{\pi }{2}$
D
$\frac{\pi }{12}$

✓

Answer

Correct option: C.

$\frac{\pi }{2}$

c
$T_{r}=\tan ^{-1}\left(\frac{\frac{1}{3}}{1+\frac{r}{3} \frac{(r-1)}{3}}\right)$

$\mathrm{T}_{\mathrm{r}}=\tan ^{-1}\left(\frac{\mathrm{r}}{3}\right)-\tan ^{-1}\left(\frac{(\mathrm{r}-1)}{3}\right)$

$\mathrm{T}_{1}=\tan ^{-1}\left(\frac{1}{3}\right)-\tan ^{-1}(0)$

$\mathrm{T}_{2}=\tan ^{-1}\left(\frac{2}{3}\right)-\tan ^{-1}\left(\frac{1}{3}\right)$

$\mathrm{T}_{\mathrm{n}}=\tan ^{-1}\left(\frac{\mathrm{n}}{3}\right)-\tan ^{-1}\left(\frac{(\mathrm{n}-1)}{3}\right)$

$S_{n}=\tan ^{-1}\left(\frac{n}{3}\right)=\frac{\pi}{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec{a}= \hat{i} + \hat{j} + \hat{k} ,\vec{b} = \hat{i} + \hat{j} , \vec{c} = \hat{i}$ and $(\vec{a} \times \vec{b}) \times \vec{c} = \lambda \vec{a} + \mu \vec{b} , then \lambda + \mu$ is equal to :-

The unit vector perpendicular to both the vectors $i -2j + 3k$ and $i + 2j -k$ is

$x - 2 = {t^2},\;y = 2t$ are the parametric equations of the parabola

Let $f$ be a function such that $f(x)\, = \,\sum\limits_{n\, - \,1}^n {\left[ {r\, + \,\cos \frac{x}{r}} \right]} $ where [.] denotes greatest integer function and $x \in [0,\pi]$, then range of $f(x)$ is-

The sum of the numbers between $100$ and $1000$, which is divisible by $9$ will be

Let $A, B, C$ are three angles such that $sinA + sinB + sinC = 0,$ then

$ \frac {sinAsin BsinC}{(sin 3A+ sin 3B+ sin 3C)}$ (wherever definied) is -

If ${A_\lambda } = \left( {\begin{array}{*{20}{c}}
\lambda &{\lambda - 1}\\
{\lambda - 1}&\lambda
\end{array}} \right);\,\lambda \in N$ then $|A_1| + |A_2| + ..... + |A_{300}|$ is equal to

Value of $\frac{{4\sin {9^o}\sin {{21}^o}\sin {{39}^o}\sin {{51}^o}\sin {{69}^o}\sin {{81}^o}}}{{\sin {{54}^o}}}$ is equal to

$\tan \left[ {2{{\tan }^{ - 1}}\left( {\frac{1}{5}} \right) - \frac{\pi }{4}} \right] = $

The solution of the differential equation $\frac{{dy}}{{dx}} + 2y\cot x = 3{x^2}{\rm{cose}}{{\rm{c}}^2}x$ is