Sets, Sequences and Series - Calculus - Statistics Previous Year Questions

1

2019 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2019

Let \(a_n = \frac{(-1)^{n+1}}{n!}, n \geq 0\), and \(b_n = \sum_{k=0}^n a_k\), \(n \geq 0\). Then, for \(|x| < 1\), the series \(\sum_{n=0}^\infty b_n x^n\) converges to

A

\(\frac{-e^{-x}}{1+x}\)

B

\(\frac{e^{-x}}{1-x^2}\)

C

\(\frac{-e^{-x}}{1-x}\)

D

\(-(1+x)e^{-x}\)

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2

2022 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2022

Let \(\{a_n\}_{n=1}^{\infty}\) be a sequence of positive real numbers satisfying
\(\frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343}\), \(n \geq 1\)
with \(a_1 = 3\) and \(a_n < 7\) for all \(n \geq 2\).
Consider the following statements:
(I) \(\{a_n\}\) is monotonically increasing.
(II) \(\{a_n\}\) converges to a value in the interval \([3,7]\).
Then which of the above statements is/are true?

A

(I) only

B

(II) only

C

Both (I) and (II)

D

Neither (I) nor (II)

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3

2022 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2022

Consider the following infinite series:
\(S_1 := \sum_{n=0}^{\infty} (-1)^n \frac{n}{n^2+4} \)   and   \(S_2 := \sum_{n=0}^{\infty} (-1)^n (\sqrt{n^2+1}-n)\).
Which of the above series is/are conditionally convergent?

A

\(S_1\) only

B

\(S_2\) only

C

Both \(S_1\) and \(S_2\)

D

Neither \(S_1\) nor \(S_2\)

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4

2023 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2023

Which of the following sets is/are countable?

A

The set of all functions from {1, 2, 3, ..., 10} to the set of all rational numbers

B

The set of all functions from the set of all natural numbers to {0, 1}

C

The set of all integer valued sequences with only finitely many non-zero terms

D

The set of all integer valued sequences converging to 1

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5

2024 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2024

Let \( \{a_n\}_{n \geq 1} \) be a sequence of real numbers such that \( a_1 = \sqrt{6} \) and \( a_{n+1} = \sqrt{6 + a_n} \) for \( n \geq 1 \). Consider the following statements: (I) \( \{a_n\}_{n \geq 1} \) is an increasing sequence. (II) \( \lim_{n \to \infty} a_n = 2 \). Which of the above statements is/are true?

A

Only (I)

B

Only (II)

C

Both (I) and (II)

D

Neither (I) nor (II)

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6

2024 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2024

Consider the power series \(\sum_{n=0}^{\infty} a_n x^n\), where \(a_{2n+1} = \frac{1}{2^{2n+1}}\) and \(a_{2n} = \frac{1}{3^{2n}}\) for \(n = 0, 1, 2, \ldots\). The radius of convergence of the power series equals __________ (in integer).

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7

2025 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2025

Among the following four statements about countability and uncountability of different sets, which is the correct statement?

A

The set \(\bigcup_{n=0}^{\infty} \{ x \in \mathbb{R} : x = \sum_{i=0}^{n} 10^i a_i \, \text{where} \, a_i \in \{1, 2\} \, \text{for} \, i = 0, 1, 2, \ldots, n \} \) is uncountable

B

The set \(\{ x \in (0,1) : x = \sum_{n=1}^{\infty} \frac{a_n}{10^n} \, \text{where} \, a_n = 1 \, \text{or} \, 2 \, \text{for each} \, n \in \mathbb{N} \} \) is uncountable

C

There exists an uncountable set whose elements are pairwise disjoint open intervals in \(\mathbb{R} \)

D

The set of all intervals with rational end points is uncountable

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8

2025 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2025

Let \(\{x_n\}_{n\geq 1}\) be a sequence defined as \(x_n = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} - 2(\sqrt{n} - 1)\).
Then which of the following options is/are correct?

A

The sequence \(\{x_n\}_{n\geq 1}\) is unbounded

B

The sequence \(\{x_n\}_{n\geq 1}\) is monotonically decreasing

C

The sequence \(\{x_n\}_{n\geq 1}\) is bounded but does not converge

D

The sequence \(\{x_n\}_{n\geq 1}\) converges

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9

2026 · Statistics · Calculus · Sets, Sequences and Series

Statistics (ST) 2026

Let \( x_1 \in (0, 4) \) and consider the sequence \( \{x_n\}_{n \geq 1} \) defined iteratively by \[ x_{n+1} = 2 - (4 - x_n)^{\frac{1}{2}}, \ n \geq 1. \] Consider the following statements: (I) \( \{x_n\} \) converges to 0 . (II) \( \left\{ \frac{x_{n+1}}{x_n} \right\} \) converges to \( \frac{1}{4} \). Which of the following statements is correct?

A

Only statement (I) is correct

B

Only statement (II) is correct

C

Both statements (I) and (II) are correct

D

Neither statement (I) nor statement (II) is correct

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