For any positive integer n, let S_{n}:(0, \infty) \to {R} be defined by s_{n}(x)=\sum_{k=1}^{n} \cot ^{-1}(\frac{1+k(k+1) x^{2}}{x}) where for any x \in{R}, \cot ^{-1} x \in(0, \pi) and \tan ^{-1}(x) \in (-\frac{\pi}{2}, \frac{\pi}{2}). Then which of the following statements is (are) TRUE? [JEE Adv 2021 P1]

S_{10}(x)=\frac{\pi}{2}-\tan ^{-1}(\frac{1+11 x^{2}}{10 x}), \text{ for all } x>0

\lim_{n \to \infty} \cot (S_{n}(x))=x, for all x>0

The equation S_{3}(x)=\frac{\pi}{4} has a root in (0, \infty)

\tan (S_{n}(x)) \leq \frac{1}{2}, for all {n} \geq 1 and x>0

Sequences and Series

Section: Mathematics

Question : 4 of 22

Marks: +1, -0

For any positive integer n, let Sn:(0,∞)→R be defined by

sn(x)=

∑

k=1

cot−1(‌

1+k(k+1)x2

)

where for any x∈R,cot−1x∈(0,π) and tan−1(x)∈ (−‌

,‌

). Then which of the following statements is (are) TRUE?

[JEE Adv 2021 P1]

Go to Question:

Prev Question

Next Question