For any positive integer n , let S n : ( 0 , ∞ ) → R be defined by S n x = ∑ k = 1 n cot - 1 1 + k ( k + 1 ) x 2 x where for any x ∈ R , cot - 1 ( x ) ∈ ( 0 , π ) and tan - 1 ( x ) ∈ - π 2 , π 2 . Then which of the following statements is (are) TRUE ?

lim n → ∞ cot S n ( x ) = x , for all x > 0

The equation S 3 ( x ) = π 4 has a root in ( 0 , ∞ )

tan S n ( x ) ≤ 1 2 , for all n ≥ 1 and x > 0

For any positive integer n , let S n : ( 0 , ∞ ) → R be defined by S n x = ∑ k = 1 n cot - 1 1 + k ( k + 1 ) x 2 x where for any x ∈ R , cot - 1 ( x ) ∈ ( 0 , π ) and tan - 1 ( x ) ∈ - π 2 , π 2 . Then which of the following statements is (are) TRUE ?

lim n → ∞ cot S n ( x ) = x , for all x > 0

HARD

JEE Advanced

IMPORTANT

Earn 100

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 $x \in R, \cot^{- 1} (x) \in (0, π)$ and $\tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2})$ . Then which of the following statements is (are) TRUE ?

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Important Questions on Sequences and Series

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2020>Q 6

Let

m

be the minimum possible value of

\log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}})

, where

y_{1}, y_{2}, y_{3}

are real numbers for which

y_{1} + y_{2} + y_{3} = 9

. Let

M

be the maximum possible value of

(\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3})

, where

x_{1}, x_{2}, x_{3}

are positive real numbers for which

x_{1} + x_{2} + x_{3} = 9

. Then the value of

\log_{2} (m^{3}) + \log_{3} (M^{2})

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2020>Q 7

Let

a_{1}, a_{2}, a_{3}, . . . . . . . . .

be a sequence of positive integers in arithmetic progression with common difference

2

. Also, let

b_{1}, b_{2}, b_{3}, . . . . . . . .

be a sequence of positive integers in geometric progression with common ratio

2

. If

a_{1} = b_{1} = c

, then the number of all possible values of

c

, for which the equality

2 (a_{1} + a_{2} + . . . . . . . . + a_{n}) = b_{1} + b_{2} + . . . . . . . . + b_{n}

holds for some positive integer

n

, is _______

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2019>Q 8

Let

A P (a; d)

denote the set of all the terms of an infinite arithmetic progression with first term

a

and common difference

d > 0 .

A P (1; 3) \cap A P (2; 5) \cap A P (3; 7) = A P (a; d)

then

a + d

equals ____

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2018>Q 9

Let

X

be the set consisting of the first

2018

terms of the arithmetic progression

1,6, 11, \dots,

and

Y

be the set consisting of the first

2018

terms of the arithmetic progression

9,16,23, \dots

. Then, the number of elements in the set

X \cup Y

is___.

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2018>Q 10

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

(- \frac{1}{2}, \frac{1}{2})

is____.

(Here, the inverse trigonometric functions

\sin^{- 1} x & \cos^{- 1} x

assume values in

[- \frac{π}{2}, \frac{π}{2}] & [0, π]

respectively.)

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2013>Q 14

The value of

\cot (Σ_{n = 1}^{23} \cot^{- 1} (1 + Σ_{k = 1}^{n} 2 k))

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 3 - Sequences and Series>JEE Advanced Paper 1 - 2013>Q 15

Let

S_{n} = Σ_{k = 1}^{4 n} {(- 1)}^{\frac{k (k + 1)}{2}} k^{2}

. Then

S_{n}

can take value(s)

For any positive integer n, let Sn:(0,∞)→R be defined by Snx=∑k=1ncot-11+k(k+1)x2x where for any x∈R,cot-1(x)∈(0,π) and tan-1(x)∈-π2,π2. Then which of the following statements is (are) TRUE ?

Important Questions on Sequences and Series

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