The Real Line

Elements of set theory: operations, De Morgan and difference free

In plain words

Every "in $A$ but not in $B$ " is really "in $A$ and outside $B$ ". Write it that way and only three operations are left: and, or, not.

Rewrite every set difference as $A \setminus B = A \cap B^{c}$ , and the whole subject collapses into Boolean algebra where De Morgan does the rest.

Exam focus: A free minute at the start of Part B if you rewrite every difference as A ∩ Bᶜ, and a two-minute case analysis if you do not. The recurring question is A \ (B \ C), and the recurring wrong answer moves the brackets.

Practice

Part B2025 Juneshow ▾

Let p, q be non-negative integers. Consider the following statements: (A) There is an integer k ≥ 1 such that p + k = q. (B) There is an integer k ≥ 1 such that q + k = p. Which of the following statements is true?

A.There exist non-negative integers p, q such that both (A) and (B) are true.
B.Both (A) and (B) are false if and only if p = q.✓ Correct
C.For all non-negative integers p and q, (A) or (B) is true.
D.There exists p ≠ q such that both (A) and (B) are false.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let S be a subset of the open interval (0,1) that consists of all the real numbers

\alpha \in (0,1)

whose infinite decimal expansion

\alpha = 0.a_{1}a_{2}a_{3}\cdots

is such that

a_{i} \in

{0,2,4} for all i ≥ 1. Which of the following statements is true?

A.There is a bijective map from $\mathbb{N}$ to S.
B.There is a surjective map from S onto (0,1).✓ Correct
C.There is a bijective map from $\mathbb{N}$ to $(0,1)\S$ .
D.S is a countable set and $(0,1)\S$ is uncountable.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Let A = {

p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}

for some

n \in \mathbb{N} \cup

{0}, gcd(p, q) = 1}, B = {

p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}5^{m}

for some

n, m \in \mathbb{N} \cup

{0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

A.A ⊊ C and B ⊊ C
B.A ⊊ C ⊊ B
C.A ⊊ B ⊊ C
D.A ⊊ B = C✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Which of the following statements is necessarily true?

A.The set of all finite subsets of $\mathbb{Z}$ is uncountable.
B.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. If $S \subseteq \mathbb{R}$ is a countably infinite set, then f(S) is a countably infinite set.✓ Correct
C.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then there exists an uncountable subset $T \subseteq \mathbb{R}$ such that $f(T) \subseteq \mathbb{R}$ is uncountable.
D.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. If R ⊆ image(f) is a countable set, then f⁻ $^{1}(R)$ is countable.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Let A, B be non-empty subsets of

\mathbb{N}

with cardinality |A| ≥ 2. Let

S_{1} =

{f : A → B | f is one-to-one} and

S_{2} =

{g : B → A | g is onto}. Which of the following statements is true?

A.If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$ .
B.If $B = \mathbb{N}$ , then there exists a one-to-one map from $S_{2}$ to B.
C.If $B = \mathbb{N}$ and A is finite, then there exists a one-to-one map from B to $S_{1}$ .✓ Correct
D.If A is finite, then $S_{2}$ is finite for any B.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let A, B, and C be sets. Which of the following sets is equal to A \ (B \ C)?

A.A \ B
B.(A \ B) ∪ C
C.A \ (B ∪ C)
D.(A \ B) ∪ (A ∩ C)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let A and B be subsets of U. Which of the following are true for all A, B?

A.A \ B = A ∩ Bᶜ for all sets A and B.✓ Correct
B.A ∪ B = U whenever A \ B = ∅.
C.If A ∩ B = ∅ then A \ B = A.✓ Correct
D.P(A) ∪ P(B) = P(A ∪ B) for all sets A and B.

Solution

(1) is the definition of set difference.

(3) holds because A ∩ B = ∅ puts A entirely inside Bᶜ, so A ∩ Bᶜ = A.

(2) is false: A \ B = ∅ only says A ⊆ B, not that B (let alone A ∪ B) is all of U — take A = ∅, B = {1}, U = {1, 2}.

(4) is false: with A = {1}, B = {2}, P(A) ∪ P(B) = {∅, {1}, {2}} but P(A ∪ B) = P({1,2}) also contains {1,2}, which is on neither side of the union.

Part Cshow ▾

Let A, B, C be subsets of a set U. Which of the following are true for all A, B, C?

A.(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ, by De Morgan's law.✓ Correct
B.A \ (B ∪ C) = (A \ B) ∩ (A \ C) for all sets.✓ Correct
C.A ∆ B = (A ∪ B) \ (A ∩ B), the symmetric difference identity.✓ Correct
D.(A ∩ B) ∪ C = A ∩ (B ∪ C) for all sets A, B, C.

Solution

(1)–(3) are standard identities: De Morgan for complements, De Morgan for relative complements, and the definition of symmetric difference — all hold for every A, B, C.

(4) is false: take A = ∅, B = ∅, C = {1}.

The left side is (∅ ∩ ∅) ∪ {1} = {1}; the right side is ∅ ∩ ({1}) = ∅.

Distributing ∩ over ∪ needs the same two sets on both occurrences of the operation — this expression doesn't have that shape.

2. Completeness, sup/inf, Archimedean property

Lec-06 Real Number System

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Builds ℝ and states the order axioms — the foundation for everything that follows.

Lec-7 LUB Axiom

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

The least-upper-bound axiom itself: this is precisely what ℚ lacks.

Lec-11 Ordered set, Least upper bound, greatest lower bound of a set

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Alternative treatment with more worked examples of sup and inf.

The real line: supremum, Archimedes and density free

In plain words

The rationals have holes in them and the reals fill every one. Completeness says any set with a ceiling has a lowest ceiling, and that lowest ceiling need not belong to the set.

The completeness axiom

Every non-empty subset of $\mathbb{R}$ bounded above has a least upper bound. This single axiom characterises $\mathbb{R}$ among ordered fields and is exactly what $\mathbb{Q}$ lacks: { $x \in \mathbb{Q} : x^{2} < 2$ } has no rational supremum.

Equivalent forms (any one implies the rest):

monotone bounded sequences converge;
Bolzano–Weierstrass;
Cauchy sequences converge;
nested interval property;
Heine–Borel on [a,b];
every non-empty bounded-below set has an infimum.

Working with sup and inf

s = sup A ⇔ s is an upper bound and for every $\varepsilon > 0$ there is a ∈ A with $a > s - \varepsilon$ .
sup(A + B) = sup A + sup B, but sup(AB) ≠ sup A · sup B in general (negatives).
sup(−A) = −inf A.
sup and inf need not be attained: sup(0,1) = 1 ∉ (0,1). They are attained for compact sets.

Archimedean property

For every $x \in \mathbb{R}$ there is $n \in \mathbb{N}$ with n > x. Consequences: 1/n → 0, and $\mathbb{Q}$ is dense in $\mathbb{R} ($ between any two reals lies a rational, and also an irrational).

Non-Archimedean ordered fields exist (fields of rational functions ordered by growth), which is why this is stated as a property, not a triviality.

Countability

$\mathbb{Q}$ is countable; $\mathbb{R}$ and the irrationals are uncountable; | $\mathbb{R}$ | = | $\mathbb{R}^{n}$ | = | $P(\mathbb{N})$ |. A countable set has measure zero, so $\mathbb{Q}$ has measure zero — but the Cantor set shows uncountable sets can too.

Key takeaways

Completeness = existence of suprema; everything else in the list is equivalent to it.
Use the $\varepsilon-$ characterisation of sup to prove sup statements.
Density of both $\mathbb{Q}$ and its complement is the source of most "nowhere continuous" examples.

What completeness of ℝ actually buys youinteractive

This visual is part of Trap Analysis

The one axiom that separates ℝ from ℚ, and the three theorems that collapse without it.

Exam focus: Every question here is really 'which property of ℝ fails in ℚ'. Completeness (sup exists) is what separates them; Archimedes and density follow from it.

Practice

Part C2025 Decembershow ▾

Consider the following subset of real numbers A = {

(1+(-1)^{n})n - 1/n : n

is a positive integer}. Which of the following statements are true?

A.A is bounded below but not bounded above.✓ Correct
B.A is bounded above but not bounded below.
C.inf A = −1✓ Correct
D.sup A = 1

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Consider the set A = {

x \in \mathbb{Q} : 0 < (\sqrt{2} - 1)x < \sqrt{2} + 1

} as a subset of

\mathbb{R}

. Which of the following statements is true?

A. $\sup A = 2 + 2\sqrt{3}$
B. $\sup A = 3 + 2\sqrt{2}$ ✓ Correct
C. $\inf A = 2 + 2\sqrt{3}$
D. $\inf A = 3 + 2\sqrt{2}$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Consider X = {u |

u : [0, 1] \to \mathbb{R}

is continuous and u(0) = 0} with the sup norm ‖u‖ = sup(x∈[0,1])|u(x)|. Let

T(u) = \int_{0}^{1}u(t)

dt and S = {|T(u)| : u ∈ X, ‖u‖ ≤ 1}. Which of the following statements are true?

A.S is an unbounded subset of $\mathbb{R}$ .
B.S is a bounded subset of $\mathbb{R}$ and sup(S) = 1.✓ Correct
C.There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.
D.S is a closed subset of $\mathbb{R}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let A, B be two non-empty subsets of

\mathbb{R}

. Let S = {

f : A \to \mathbb{R}

| f is continuous}, T = {

f : B \to \mathbb{R}

| f is continuous}, U = {

f : A \times A \to \mathbb{R}

| f is continuous}. Which of the following statements is true?

A.If A is finite, then there exists a bijection between S and U.✓ Correct
B.If A is finite and B = [0, 1], then there is no bijection between S and T.
C.There is no bijection between S and U for any choice of A.
D.If A ≠ B, then there is no bijection between T and U.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let S = {

x \in \mathbb{R} : x > 1

and

(1 - x^{4})/(1 - x^{3}) > 22

}. Which of the following is true about S?

A.S is empty.
B.There is a bijection between S and $\mathbb{N}$
C.There is a bijection between S and $\mathbb{R}$ ✓ Correct
D.There is a bijection between S and a non-empty finite set

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true?

A.There exists a non-empty finite set in C
B.There exists a countably infinite set in C
C.There exists an uncountable set in C
D.C is empty✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For a real number x, let [x] denote the largest integer ≤ x. Which of the following sets are uncountable?

A.{ $x \in \mathbb{R}$ | [x] = 1}✓ Correct
B.{ $x \in \mathbb{R}$ | x − [x] = 1/2}
C.{ $x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$ }
D.{ $x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$ }

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Which of the following statements are true?

A.Let $x, y \in \mathbb{R}$ with x < y. Then there exists $r \in \mathbb{Q}$ such that x < (2^2024/e)r < y.✓ Correct
B.Let $(a_{n})_{n}\ge_{2}$ be a sequence of positive real numbers. If there exists a positive real number L such that $\lim \sup(n\to\infty) a_{n}/\log n = L$ , then $\lim \sup(n\to\infty) a_{n} < \infty$ .
C.The set of all finite subsets of $\mathbb{Q}$ is countably infinite.✓ Correct
D.The set of continuous functions from $\mathbb{R}$ to the set {0, 1} is infinite.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

A, B \subset \mathbb{R}

be non-empty and bounded. Which of the following are true?

A.sup(A ∪ B) = max{sup A, sup B}✓ Correct
B.sup(A + B) = sup A + sup B✓ Correct
C.sup(A ∩ B) = min{sup A, sup B} whenever A ∩ B ≠ ∅
D.inf A ≤ sup A✓ Correct

Solution

(c) fails: A = [0,1], B = [0,2] ∪ {3} … more simply A = {0, 2}, B = {0, 3} gives A ∩ B = {0} with sup 0, while min{2,3} = 2.

Part Bshow ▾

Which of the following subsets of

\mathbb{R}

has a supremum that is not attained?

A.[0, 1]
B.{ $1 - 1/n : n \in \mathbb{N}$ }✓ Correct
C.{0, 1}
D.a finite set

The worked solution and the per-option analysis are in Trap Analysis.

3. Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Lec-08 Sequences of Real Numbers

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Convergence, boundedness and the monotone convergence theorem.

Lec-18 Fundamental theorems on limits, Bolzano-Weierstrass Theorem

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Bolzano–Weierstrass in full — the workhorse of every 'bounded sequence' question.

Lec-20 Cauchy sequence and its properties

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Cauchy criterion; note where completeness of ℝ is used.

Sequences in ℝ — the implication map free

In plain words

Converging means the terms settle on a number. Cauchy means they settle near each other without naming where. In the reals these agree, because there is no hole to crowd around and fall into.

The four properties and how they relate

For a real sequence $(a_{n})$ :

convergent ──▶ Cauchy ──▶ bounded ──▶ has a convergent subsequence (Bolzano–Weierstrass)
     ▲                                          │
     └────────── (in ℝ, Cauchy ⇒ convergent) ◀──┘  only if the subsequential limit is unique

Implication	Holds?	Why / spoiler
convergent ⇒ Cauchy	✓ always	triangle inequality
Cauchy ⇒ convergent	✓ in $\mathbb{R} ($ completeness)	fails in $\mathbb{Q}$ : decimal approximations of $\sqrt{2}$
Cauchy ⇒ bounded	✓ always	finitely many terms outside an $\varepsilon-$ ball
bounded ⇒ convergent	✗	$(-1)^{n}$
bounded ⇒ convergent subsequence	✓ in $\mathbb{R} (B$ –W)	fails in infinite-dimensional spaces
monotone + bounded ⇒ convergent	✓ (MCT)	the workhorse for recursively defined sequences

Standard exam sequences

$a_{n} = (1 + 1/n)^{n}$ — increasing, bounded by 3, limit e.
** $a_{n} =$ $n^{1/n}**$ — → 1. (Write $n^{1/n}$ $= 1 + h_{n}$ , expand with binomial: $n \ge 1 + n(n-1)h_{n}^{2}/2.)$
** $a_{n} = (n$ ! $)^{1/n}/n**$ — → 1/e (Stirling or Cauchy's root/ratio lemma).
$a_{n}$ ₊ $_{1} = \sqrt{2 + a_{n}}, a_{1} = \sqrt{2}$ — increasing, bounded by 2, limit 2. Recursive sequences: prove monotone + bounded by induction, then solve L = f(L).
$a_{n} = \sin n$ — bounded, divergent, set of subsequential limits is all of [−1, 1] (density of { $n mod 2\pi$ }).

Cauchy's lemmas (quote them)

If $a_{n} \to L$ then $(a_{1} + \cdots + a_{n})/n \to L ($ Cesàro). Converse fails: $a_{n} = (-1)^{n}$ .
If $a_{n} > 0$ and $a_{n}$ ₊ $_{1}/a_{n} \to L$ then $a_{n}^{1/n}$ → L. Converse fails: 1, 2, 1, 2, 1, 2, … raised appropriately.

One-line disproof technique

To show "bounded ⇒ X" fails, reach for $(-1)^{n}$ . To show "Cauchy ⇒ convergent" fails outside $\mathbb{R}$ , reach for $\mathbb{Q}$ or (0, 1) with 1/n. Examiners reuse these three spoilers constantly.

Key takeaways

In $\mathbb{R}$ : convergent ⇔ Cauchy; bounded ⇒ convergent subsequence; monotone + bounded ⇒ convergent.
Uniqueness of subsequential limit + boundedness ⇒ convergence.
Cesàro means and the ratio→root lemma go one way only.

Bolzano–Weierstrass: the bisection huntinteractive

This visual is part of Trap Analysis

sin n never converges — bisect its range and watch nested intervals trap a convergent subsequence anyway.

Exam focus: Bolzano–Weierstrass and the Cauchy criterion are tested as decisions: given a sequence, is it bounded / Cauchy / convergent / does it have a convergent subsequence? Know exactly which implications hold in ℝ and which need completeness.

Practice

Part B2025 Juneshow ▾

Let

f : \mathbb{R}

\mathbb{Q} \to \mathbb{R}

\mathbb{Q}

be the function defined as f(x) = (3x + 2)/(4x + 3). Let

x_{1} \in \mathbb{R}

\mathbb{Q}

. For n ≥ 1, define

x_{n}

₊

_{1} = f(x_{n})

. Suppose that the sequence

(x_{n})_{n}\ge_{1}

converges to a real number

\ell

. Which of the following statements is true?

A.If $\ell$ is positive, then $\ell = \sqrt{3}/2$ .
B.If $\ell$ is positive, then $\ell = 1/\sqrt{2}$ .✓ Correct
C.If $\ell$ is negative, then $\ell = -\sqrt{3}/2$ .
D.If $\ell$ is negative, then $\ell = -1/2$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let

\alpha = \sum_{l=1}^{2025}

a_{l}/10

ˡ with

a_{l} \in

{0,1,…,9} for all 1≤l≤2025 and

a_{2025}\ne0

. Let

(\beta_{n})_{n}\ge1

be a strictly increasing sequence of positive real numbers in (0,1) that converges to

\alpha

. For each n≥1, write

\beta_{n} = \sum_{l=1}^\infty

b_{n,l}/10

ˡ with

b_{n,l}

∈{0,1,…,9}, for the infinite decimal expansion of

\beta_{n}

. Which of the following statements are necessarily true?

A.There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.✓ Correct
B.There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$ .✓ Correct
C. $\beta_{n}$ is rational for infinitely many n.
D.There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

For a non-negative real number a, let

\sqrt{a}

denote its non-negative square-root. Consider the function

f : \mathbb{R} \to \mathbb{R}

given by

f(x) = x(\sqrt{x^{2}+3} - \sqrt{x^{2}+2})

. Which of the following statements is true?

A. $\lim (x\to\infty) f(x) = \infty$
B. $\lim (x\to\infty) f(x) = 0$
C. $\lim (x\to\infty) f(x) = 1$
D. $\lim (x\to\infty) f(x) = 1/2$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Consider the sequences

(a_{n})_{n}\ge_{1}

and

(b_{n})_{n}\ge_{1}

, defined by

a_{n} = -2\sqrt{n} + \sum(k=1..n) 1/\sqrt{k}

and

b_{n} = -2\sqrt{n+1} + \sum(k=1..n) 1/\sqrt{k}

. Which of the following statements is true?

A. $(a_{n})_{n}\ge_{1}$ converges but $(b_{n})_{n}\ge_{1}$ does not converge.
B. $(b_{n})_{n}\ge_{1}$ converges but $(a_{n})_{n}\ge_{1}$ does not converge.
C.Both $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ converge.✓ Correct
D.Neither $(a_{n})_{n}\ge_{1}$ nor $(b_{n})_{n}\ge_{1}$ converges.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let A = {

1/(\sqrt{n + \sqrt{n}} - \sqrt{n})

n \in \mathbb{Z}, n > 0

}. Which of the following statements are true?

A.sup A is finite.✓ Correct
B. $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A$ ✓ Correct
C. $\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2$ ✓ Correct
D. $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let

(a_{n})_{n}\ge1

be a bounded sequence in

\mathbb{R}

. Which of the following statements is FALSE?

A.if $liminf_{n}\to\infty a_{n} = limsup_{n}\to\infty a_{n}$ , then $(a_{n})$ is convergent
B.if inf{ $a_{n}$ | n ≥ 1} $= limsup_{n}\to\infty a_{n}$ , then $(a_{n})$ is convergent
C.if sup{ $a_{n}$ | n ≥ 1} $= liminf_{n}\to\infty a_{n}$ , then $(a_{n})$ is constant✓ Correct
D.if sup{ $a_{n}$ | n ≥ 1} = inf{ $a_{n}$ | n ≥ 1}, then $(a_{n})$ is constant

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

(a_{n})_{n}\ge_{1}

be a sequence of positive real numbers. Let

b_{n} = a_{n} / \max

{

a_{1}, \dots, a_{n}

}, n ≥ 1. Which of the following statements are necessarily true?

A.If $lim_{n}\to\infty b_{n}$ exists in $\mathbb{R}$ , then { $a_{n} : n \ge 1$ } is bounded
B.If $lim_{n}\to\infty b_{n} = 1$ , then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$
C.If $lim_{n}\to\infty b_{n} = 1/2$ , then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$ ✓ Correct
D.If $lim_{n}\to\infty b_{n} = 0$ , then $lim_{n}\to\infty a_{n} = 0$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Consider the sequences

(a_{n})_{n}\ge_{1}

and

(b_{n})_{n}\ge_{1}

defined by

a_{n} = (e^{n} + e

⁻

^{n})/2

and

b_{n} = a_{n}

₊

_{1}/a_{n}

. Which of the following statements is true?

A.For every $x \in \mathbb{R}$ there exists some n such that $a_{n} > x$ ✓ Correct
B.For every $x \in \mathbb{R}$ there exists some n such that $a_{n} < x$
C.For every $x \in \mathbb{R}$ there exists some n such that $b_{n} > x$
D.For every $x \in \mathbb{R}$ there exists some n such that $b_{n} < x$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾

Under which of the following conditions is the sequence {

x_{n}

} of real numbers convergent?

A.The subsequences { $x_{2n}$ ₊ $_{1}$ }, { $x_{2n}$ } and { $x_{3n}$ } are convergent and have the same limit.✓ Correct
B.The subsequences { $x_{2n}$ ₊ $_{1}$ }, { $x_{2n}$ } and { $x_{3n}$ } are convergent.✓ Correct
C.The subsequences { $x_{kn}$ } $_{n}$ are convergent for every k ≥ 2.
D.lim | $x_{n}$ ₊ $_{1} - x_{n}$ | = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Consider the following subset of

\mathbb{R}: U =

{

x \in \mathbb{R} : x^{2} - 9x + 18 \le 0, x^{2} - 7x + 12 \le 0

}. Which one of the following statements is true?

A.inf U = 5.
B.inf U = 4.
C.inf U = 3.✓ Correct
D.inf U = 2.

Solution

$x^{2} - 9x + 18 \le 0 \Leftrightarrow x \in [3, 6]; x^{2} - 7x + 12 \le 0 \Leftrightarrow x \in [3, 4]$ .

So U = [3, 4] and inf U = 3.

Part C2023 Juneshow ▾

Let

f : \mathbb{R} \to \mathbb{R}

be defined as

f(x) = 1/4 + x - x^{2}

. Given

a \in \mathbb{R}

, define the sequence {

x_{n}

} by

x_{0} = a

and

x_{n} = f(x_{n}

₋

_{1})

for n ≥ 1. Which of the following statements are true?

A.If a = 0, then the sequence { $x_{n}$ } converges to 1/2.✓ Correct
B.If a = 0, then the sequence { $x_{n}$ } converges to −1/2.
C.The sequence { $x_{n}$ } converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.✓ Correct
D.If a = 0, then the sequence { $x_{n}$ } does not converge.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Let x be a real number. Which of the following statements are true?

A.There exists an integer n ≥ 1 such that $n^{2} \sin(1/n) \ge x$ .✓ Correct
B.There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.✓ Correct
C.There exists an integer n ≥ 1 such that n $e^{-n}$ ≥ x.
D.There exists an integer n ≥ 2 such that n (log $n)^{-1}$ ≥ x.✓ Correct

Solution

$n^{2} \sin(1/n)$ ~ n, n cos(1/n) ~ n and n/log n all tend to $+\infty$ , so each eventually exceeds any x. n $e^{-n}$ ≤ 1/e is bounded, so it fails for x > 1/e.

Part Bshow ▾

The limit of the sequence

a_{n} =

n^{1/n}

n \to \infty

A.0
B.1✓ Correct
C.e
D.does not exist

Solution

Write $n^{1/n}$ = $e^{(\log n)/n}$ and (log n)/n → 0, so the limit is $e^{0} = 1$ .

Alternatively set $a_{n} = 1 + h_{n}$ and expand $(1 + h_{n})^{n} \ge C(n,2)h_{n}^{2}$ to force $h_{n} \to 0$ .

Part Cshow ▾

Let

(a_{n})

be a sequence of real numbers. Which of the following are true?

A.If $(a_{n})$ is Cauchy then it is bounded✓ Correct
B.If $(a_{n})$ is bounded then it has a convergent subsequence✓ Correct
C.If every subsequence of $(a_{n})$ has a further subsequence converging to L, then $a_{n} \to L$ ✓ Correct
D.If $(a_{1} + \cdots + a_{n})/n$ converges then $(a_{n})$ converges

Solution

(a), (b) standard.

(d) false: $a_{n} = (-1)^{n}$ has Cesàro means → 0 but diverges.

Part Bshow ▾

The set of subsequential limits of the sequence

a_{n} = \sin n

A.{0}
B.{−1, 1}
C.[−1, 1]✓ Correct
D.empty

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let

a_{1} = \sqrt{2}

and

a_{n}

₊

_{1} = \sqrt{2 + a_{n}}

. Then

(a_{n})

A.is increasing and converges to 2✓ Correct
B.is decreasing and converges to $\sqrt{2}$
C.is unbounded
D.converges to $1 + \sqrt{2}$

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

a_{n} > 0

for all n. Which of the following are true?

A.If $a_{n}$ ₊ $_{1}/a_{n} \to L$ then $a_{n}^{1/n}$ → L✓ Correct
B.If $a_{n}^{1/n}$ → L then $a_{n}$ ₊ $_{1}/a_{n} \to L$
C.lim (n! $)^{1/n}/n$ = 1/e✓ Correct
D.lim $n^{1/n}$ = 1✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let {

a_{n}

} be a sequence of real numbers. Which of the following statements are true?

A.If $\sum$ | $a_{n}$ ₊ $_{1} - a_{n}$ | converges, then { $a_{n}$ } converges.✓ Correct
B.If | $a_{n}$ ₊ $_{1} - a_{n}$ | → 0, then { $a_{n}$ } is Cauchy.
C.If { $a_{n}$ } is Cauchy and has a convergent subsequence, then { $a_{n}$ } converges.✓ Correct
D.Every bounded sequence is Cauchy.

Solution

(1) The partial sums of $\sum(a_{n}$ ₊ $_{1} - a_{n})$ form a telescoping series that converges absolutely, so { $a_{n}$ } converges.

(2) Fails: $a_{n} = \sqrt{n}$ has | $a_{n}$ ₊ $_{1} - a_{n}$ | → 0 but is unbounded; the partial sums of the harmonic series are a bounded counterexample.

(3) True, though the subsequence hypothesis is redundant — a Cauchy sequence in $\mathbb{R}$ converges anyway.

(4) Fails: $a_{n} = (-1)^{n}$ is bounded and not Cauchy.

4. limsup, liminf and subsequential limits

Lec-09 Sequences of Real Numbers - continued

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Subsequences and subsequential limits, leading into limsup/liminf.

Lec-17 Cauchy theorems on limit of sequences with examples

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Includes the Cesàro/ratio-root lemmas that the exam quotes.

limsup and liminf — three equivalent definitions and the algebra free

In plain words

A sequence that never settles still has a highest and a lowest level it keeps returning to. Those two are the limsup and the liminf, and they exist even when the limit does not.

Definitions (all equivalent for real sequences)

Tail sup/inf: $\limsup a_{n} =$ $lim_{n\to\infty}$ $sup_{k\ge{}n}$ $a_{k} = inf_n$ $sup_{k\ge{}n}$ $a_{k} ($ decreasing limit). liminf similarly.
Largest/smallest subsequential limit (in the extended reals).
$\varepsilon-$ characterisation: $L = \limsup a_{n} \Leftrightarrow$ for every $\varepsilon > 0, a_{n} < L + \varepsilon$ eventually and $a_{n} > L - \varepsilon$ infinitely often.

Definition 3 is what most Part-C statements are really testing.

Facts to quote

$- \liminf a_{n} \le \limsup a_{n}$ ; equality (finite $) \Leftrightarrow a_{n}$ converges. $- \limsup(-a_{n}) = -\liminf a_{n}$ . $- \limsup(a_{n} + b_{n}) \le \limsup a_{n} + \limsup b_{n}$ . Strict is possible: $a_{n} = (-1)^{n}, b_{n} = (-1)^{n}$ ⁺ $^{1}$ gives 0 < 2.

If $b_{n} \to b$ then $\limsup(a_{n} + b_{n}) = \limsup a_{n} + b$ .
For $a_{n}, b_{n} \ge 0: \limsup(a_{n}b_{n}) \le \limsup a_{n} \cdot \limsup b_{n}$ . $- \limsup a_{n} = +\infty \Leftrightarrow (a_{n})$ is unbounded above.

Worked computations

Sequence	limsup	liminf
$(-1)^{n}(1 + 1/n)$	1	−1
$\sin(n\pi/2)$	1	−1
$n \sin(n\pi/2)$	$+\infty$	$-\infty$
$\cos(n\pi/3)$	1	−1 (period 6: 1, ½, −½, −1, −½, ½)
$(1 + (-1)^{n}/n)^{n}$	e	1/e
{ $n\sqrt{2}$ } (fractional part)	1	0

Root test in limsup form

$\sum{}a_{n}$ converges absolutely if limsup | $a_{n}|^{1/n}$ < 1 and diverges if > 1. The limsup form is always applicable (the limit may not exist) — a frequent Part-B point.

Key takeaways

Use the $\varepsilon-$ characterisation to verify a claimed limsup.
Subadditivity of limsup can be strict; equality needs one sequence convergent.
limsup of a periodic pattern = max of the cycle; liminf = min.

Computing limsup and liminf without guessinginteractive

This visual is part of Trap Analysis

A worked sequence, the subsequential limit set, and the inequality that is only ever one-way.

Exam focus: Compute limsup/liminf of explicit sequences fast, and know the algebra: limsup(aₙ + bₙ) ≤ limsup aₙ + limsup bₙ, with equality if one converges. Convergence ⇔ limsup = liminf (finite).

Practice

Part B2025 Decembershow ▾

For a non-negative real number a, let

\sqrt{a}

denote its non-negative square-root. Consider the sequence {

x_{n}

}

_{n}\ge_{1}

defined by

x_{1} = 1

and

x_{n}

₊

_{1} = \sqrt{1 + x_{n}^{2}/n}

for n ≥ 1. Which of the following statements is true?

A. $\limsup x_{n} = 1$ ✓ Correct
B. $\limsup x_{n} = \sqrt{2}$
C. $\limsup x_{n} = 2$
D. $\limsup x_{n} = 1 + \sqrt{2}$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Consider the sequences

(s_{n})_{n}\ge_{1}

and

(t_{n})_{n}\ge_{1}

defined by

s_{n} = \sum_{k=0}^{n}

1/(k!)^{2}

and

t_{n} = \sum_{k=0}^{n}

C(n, k)(−1)ᵏ/nᵏ. Which of the following statements are true?

A. $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$ ✓ Correct
B. $limsup_{n\to\infty}$ $t_{n} \le e$ ✓ Correct
C. $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$
D. $liminf_{n\to\infty}$ $t_{n} \ge e$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

(x_{n})_{n}\ge_{1}

be a sequence of real numbers that has a decreasing subsequence

(x_(n_{k}))_{k}\ge_{1}

. Assume that

\lim(k\to\infty) x_(n_{k}) = 2025

. Which of the following statements is necessarily true?

A. $\lim \inf(n\to\infty) x_{n} \ge 2025$
B. $\lim \sup(n\to\infty) x_{n} \le 2025$
C. $\lim \inf(n\to\infty) x_{n} \le 2025$ ✓ Correct
D. $\lim \inf(k\to\infty) x_(n_{k}) > 2025$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

(a_{n})_{n}\ge_{1}

be a bounded sequence of real numbers such that

lim_{n}\to\infty a_{n}

does not exist. Let S = {

l \in \mathbb{R}

: there exists a subsequence of

(a_{n})

converging to l}. Which of the following statements are necessarily true?

A.S is the empty set
B.S has exactly one element
C.S has at least two elements✓ Correct
D.S has to be a finite set

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For a positive integer n and a subset S of the set of positive integers, let S(n) denote the set {s ∈ S | s ≤ n}. Let X be a subset of the set of positive integers such that

\lim(n\to\infty)

|X(n)|/n = 1. Assume that there exist pairwise disjoint subsets

X_{1}, X_{2}, \dots, X_{8}

of X such that ⋃

(i=1..8) X_{i} = X

. Which of the following statements are true?

A. $\lim(n\to\infty)$ | $X_{i}(n)$ |/n exists for all 1 ≤ i ≤ 8.
B. $\lim \inf(n\to\infty)$ | $X_{i}(n)$ |/n ≥ 0 for all 1 ≤ i ≤ 8.✓ Correct
C. $\lim \sup(n\to\infty)$ | $X_{i}(n)$ |/n ≥ 1/8 for some 1 ≤ i ≤ 8.✓ Correct
D. $\lim \sup(n\to\infty)$ | $X_{i}(n)$ |/n < 1/8 for all 1 ≤ i ≤ 8.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

(a_{n})_{n}\ge_{1}, (b_{n})_{n}\ge_{1}

and

(c_{n})_{n}\ge_{1}

be sequences given by

a_{n} = (-1)^{n}(1 + e

⁻

^{n}), b_{n} = \max

{

a_{1}, \dots, a_{n}

}, and

c_{n} = \min

{

a_{1}, \dots, a_{n}

}. Which of the following statements are true?

A. $(a_{n})_{n}\ge_{1}$ does not converge.✓ Correct
B. $\lim \sup(n\to\infty) a_{n} = \lim(n\to\infty) b_{n}$
C. $\lim \inf(n\to\infty) a_{n} = \lim(n\to\infty) c_{n}$
D. $\lim(n\to\infty) b_{n} = \lim(n\to\infty) c_{n}$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Which of the following assertions is correct?

A. $limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1$ .
B. $lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.
C. $liminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi$ .✓ Correct
D. $lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Consider the sequence

(a_{n})_{n\ge1}

, where

a_{n} = \cos((-1)^{n} n\pi/2 + n\pi/3)

. Which one of the following statements is true?

A. $\limsup a_{n} = \sqrt{3}/2$ .
B. $\limsup a_{2n} = 1$ .✓ Correct
C. $\limsup a_{2n} = 1/2$ .
D. $\limsup a_{3n} = 0$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

limsup of the sequence

\cos(n\pi/3)

equals

A.1✓ Correct
B.½
C.0
D.−1

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

For

a_{n} = (1 + (-1)^{n}/n)^{n}

, the pair

(\limsup a_{n}, \liminf a_{n})

A.(e, 1/e)✓ Correct
B.(e, e)
C.(1, 1)
D.(e, 0)

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

(a_{n}), (b_{n})

be bounded real sequences. Which of the following are true?

A. $\limsup(a_{n} + b_{n}) \le \limsup a_{n} + \limsup b_{n}$ ✓ Correct
B. $\limsup(a_{n} + b_{n}) = \limsup a_{n} + \limsup b_{n}$
C.If $b_{n}$ converges then $\limsup(a_{n} + b_{n}) = \limsup a_{n} + \lim b_{n}$ ✓ Correct
D. $\liminf a_{n} = -\limsup(-a_{n})$ ✓ Correct

Solution

(b) fails for $a_{n} = (-1)^{n}, b_{n} = (-1)^{n}$ ⁺ $^{1}$ : LHS 0, RHS 2.

Part Cshow ▾

Let

a_{n} = (-1)^{n}(1 + 1/n)

for n ≥ 1. Which of the following are true?

A. $\limsup a_{n} = 1$ .✓ Correct
B. $\liminf a_{n} = -1$ .✓ Correct
C.The set of subsequential limits of { $a_{n}$ } is {−1, 1}.✓ Correct
D.{ $a_{n}$ } converges.

Solution

Even-indexed terms decrease to 1, odd-indexed terms increase to −1, and every subsequence eventually follows one of these two tracks, so the subsequential limit set is exactly {−1, 1}; limsup is its largest element 1, liminf its smallest, −1.

The sequence diverges because limsup ≠ liminf.

Part Bshow ▾

Let

a_{n} = (-1)^{n} (1 + 1/n)

. Then

\limsup a_{n}

and

\liminf a_{n}

are

A.1 and −1✓ Correct
B.2 and −2
C.1 and 0
D.0 and −1

The worked solution and the per-option analysis are in Trap Analysis.

5. Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Lec-11 Infinite Series of Real Numbers

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

Definitions and the basic convergence tests.

Lec-22 Comparison tests for series, Absolutely and Conditionally convergent

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

The absolute-vs-conditional distinction the exam leans on.

Lec-24 Raabe's test, limit of functions, Cluster point

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Raabe's test — what to reach for when the ratio test gives 1.

Series tests — decision tree and the trap list free

In plain words

Terms shrinking to zero is necessary and never sufficient. Absolute convergence is the strong kind that survives reordering. The weak kind lives on cancellation, so shuffling the terms changes the answer.

Decision tree

$a_{n}$ ↛ 0? → diverges. (Converse fails: $\sum1/n.)$
Terms positive?
- looks like a p-series / geometric → comparison / limit comparison $(\sum1/n$ ᵖ converges iff p > 1).
- factorials or $n^{n} \to$ ratio test $(\lim a_{n}$ ₊ $_{1}/a_{n} = L$ : converge if L < 1, diverge if L > 1, inconclusive if L = 1).
- n-th powers → root test $(\limsup a_{n}^{1/n})$ ; root test is stronger: whenever ratio works, root works.
- ratio gives 1 → Raabe: $n(a_{n}/a_{n}$ ₊ $_{1} - 1) \to R$ : converge if R > 1, diverge if R < 1. $- a_{n}$ decreasing → Cauchy condensation: $\sum{}a_{n}$ ~ $\sum2^{k}a_{2^{k}}$ . Gives $\sum1/(n \log n)$ diverges, $\sum1/(n \log^{2} n)$ converges. $- a_{n} = f(n), f$ decreasing → integral test.
Alternating? → Leibniz: $a_{n}$ ↓ $0 \Rightarrow \sum(-1)^{n}a_{n}$ converges. Check absolute convergence separately.
$\sum{}a_{n}b_{n}$ with $b_{n}$ monotone bounded? → Abel / Dirichlet $(e.g. \sum \sin(n)/n$ converges).

The trap list

Claim	Truth	Spoiler
$a_{n} \to 0 \Rightarrow \sum{}a_{n}$ converges	✗	$\sum1/n$
$\sum{}a_{n}$ converges $\Rightarrow \sum{}a_{n}^{2}$ converges	✗	$a_{n} = (-1)^{n}/\sqrt{n} ($ true if $a_{n} \ge 0)$
$\sum{}a_{n}^{2}$ converges $\Rightarrow \sum{}a_{n}$ converges	✗	$a_{n} = 1/n$
$\sum{}a_{n}, \sum{}b_{n}$ converge $\Rightarrow \sum{}a_{n}b_{n}$ converges	✗	$a_{n} = b_{n} = (-1)^{n}/\sqrt{n}$
Convergent series can be rearranged freely	✗ unless absolutely convergent	Riemann: a conditionally convergent series can be rearranged to any sum, or to diverge
Ratio test inconclusive ⇒ root test inconclusive	✗	$a_{n} =$ $2^{-n + (-1)^{n}}$ : ratio oscillates (½·4, ½·¼), root → ½
$\sum{}a_{n}$ converges and $a_{n} \ge 0$ decreasing $\Rightarrow n\cdot{}a_{n} \to 0$	✓ (Pringsheim / Olivier)	needed for some Part-C items

Standard values

$\sum1/n^{2} = \pi^{2}/6 \cdot \sum(-1)^{n}$ ⁺ $^{1}/n = \ln 2 \cdot \sum1/n(n+1) = 1 \cdot \sum{}x^{n}/n! = e$ ˣ $\cdot \sum(-1)^{n}/(2n+1) = \pi/4$

Key takeaways

Root ≥ ratio in power; Raabe rescues ratio = 1; condensation handles logs.
Absolute convergence is what lets you rearrange, multiply, and regroup. $- (-1)^{n}/\sqrt{n}$ is the universal spoiler for "conditionally convergent but …".

Choosing a convergence test without wasting timeinteractive

This visual is part of Trap Analysis

Which test to reach for, in what order, and the family where ratio and root both return 1.

Exam focus: Know which test to reach for in 10 seconds, the limsup forms, and the three classic traps: conditional vs absolute convergence, rearrangements (Riemann), and 'terms → 0 does not imply convergence'.

Practice

Part B2025 Decembershow ▾

Let

\alpha, \beta \in (0, \infty)

. Consider the infinite series

\sum_{n\ge4}

1/[n(log_{e} n)^\alpha (log_{e}(log_{e} n))^\beta]

. Which of the following statements is true?

A.The series converges for all $\alpha \in (0,1)$ and for all $\beta \in (1,\infty)$ .
B.The series converges for $\alpha = 1$ and $\beta = 1$ .
C.The series converges for $\alpha = 1$ and for all $\beta \in (1,\infty)$ .✓ Correct
D.The series converges for all $\alpha \in (0,\infty)$ and for all $\beta \in (0,\infty)$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

\sum_{n}

₌

_{1}^\infty a_{n}

be a convergent series of real numbers. For n ≥ 1 define

A_{n} = a_{n}

a_{n} > 0

and 0 otherwise;

B_{n} = a_{n}

a_{n} < 0

and 0 otherwise. Which of the following statements are necessarily true?

A. $A_{n} \to 0$ and $B_{n} \to 0$ as $n \to \infty$ ✓ Correct
B.If $\sum_{n}$ ₌ $_{1}^\infty a_{n}$ is absolutely convergent, then both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are absolutely convergent✓ Correct
C.Both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are convergent
D.If $\sum_{n}$ ₌ $_{1}^\infty a_{n}$ is not absolutely convergent, then both $\sum_{n}$ ₌ $_{1}^\infty A_{n}$ and $\sum_{n}$ ₌ $_{1}^\infty B_{n}$ are divergent✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Consider the series

\sum_{n=1}^\infty

a_{n}

, where

a_{n} =

(-1)^{n+1}(\sqrt{n+1} - \sqrt{n})

. Which of the following statements is true?

A.The series is divergent.
B.The series is convergent.
C.The series is conditionally convergent.✓ Correct
D.The series is absolutely convergent.

Solution

$\sqrt{n+1} - \sqrt{n} = 1/(\sqrt{n+1} + \sqrt{n})$ decreases to 0, so the alternating series converges (Leibniz).

The absolute series behaves like $\sum 1/(2\sqrt{n})$ , which diverges.

Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

Part B2023 Decembershow ▾

Consider the following infinite series:

(a) \sum_{n\ge1}

\sin(n\pi/2)/\sqrt{n}, (b) \sum_{n\ge1}

\log(1 + 1/n^{2})

. Which one of the following statements is true?

A.(a) is convergent, but (b) is not convergent.
B.(a) is not convergent, but (b) is convergent.
C.Both (a) and (b) are convergent.✓ Correct
D.Neither (a) nor (b) is convergent.

Solution

(a): the non-zero terms are $(-1)^{k}/\sqrt{2k+1}$ , an alternating series with terms decreasing to 0 (Leibniz).

$(b): \log(1 + 1/n^{2})$ ~ $1/n^{2}$ , comparison with a convergent p-series.

Part Bshow ▾

The series

\sum (-1)^{n} n/(n + 1)

A.converges absolutely
B.converges conditionally
C.diverges✓ Correct
D.converges to 0

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Which of the following series converges?

A. $\sum_{n\ge2}$ 1/(n log n)
B. $\sum_{n\ge2}$ $1/(n (\log n)^{2})$ ✓ Correct
C. $\sum_{n\ge1}$ $1/\sqrt{n}$
D. $\sum_{n\ge1}$ $n/(n^{2} + 1)$

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

\sum{}a_{n}

be a convergent series of real numbers. Which of the following are necessarily true?

A. $\sum{}a_{n}^{2}$ converges
B.If $a_{n} \ge 0$ then $\sum{}a_{n}^{2}$ converges✓ Correct
C.Every rearrangement of $\sum{}a_{n}$ converges to the same sum
D. $a_{n} \to 0$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

a_{n} = 1/n

ᵖ for n ≥ 1 and a real parameter p > 0. Which of the following are true?

A. $\sum{}a_{n}$ converges if and only if p > 1.✓ Correct
B.The ratio test is inconclusive for $\sum{}a_{n}$ for every value of p.✓ Correct
C.The root test is also inconclusive for $\sum{}a_{n}$ for every value of p.✓ Correct
D.Raabe's test can never determine convergence of $\sum{}a_{n}$ , since the ratio test already fails there.

Solution

(1) is the p-series test.

$(2) a_{n}$ ₊ $_{1}/a_{n} = (n/(n+1))$ ᵖ → 1 regardless of p, so the ratio test limit is always 1.

$(3) (a_{n})^{1/n}$ = $n^{-p/n}$ → 1 likewise.

(4) is false, and it is exactly backwards: Raabe's test exists precisely to resolve the cases the ratio test cannot.

Here $n(1 - a_{n}$ ₊ $_{1}/a_{n}) \to p$ , so Raabe's test recovers the correct threshold — converges for p > 1, diverges for p < 1 — using the very case the ratio test left undecided.

Part Cshow ▾

Which of the following are true?

A. $\sum_{n=2}^\infty$ 1/(n log n) diverges, by the Cauchy condensation test.✓ Correct
B. $\sum_{n=1}^\infty$ $(-1)^{n}/n$ converges absolutely.
C.If $\sum{}a_{n}$ converges absolutely, then every rearrangement of $\sum{}a_{n}$ converges to the same sum.✓ Correct
D.By Raabe's test, if $\lim n(1 - a_{n}$ ₊ $_{1}/a_{n}) = L > 1$ for a positive-term series $\sum{}a_{n}$ , then $\sum{}a_{n}$ converges.✓ Correct

Solution

(1) condensation reduces it to $\sum2^{n}/(2^{n}\cdot{}n \log 2) = (1/\log 2)\sum1/n$ , divergent.

(3) is the rearrangement theorem for absolutely convergent series.

(4) is the statement of Raabe's test.

(2) is false: $\sum(-1)^{n}/n$ is the alternating harmonic series, convergent by Leibniz's test but only conditionally — $\sum1/n$ diverges, so it is not absolutely convergent.

Part Bshow ▾

For the series

\sum n!/n^{n}

, the ratio test gives the limit

A.1/e, so the series converges✓ Correct
B.e, so the series diverges
C.1, inconclusive
D.0, so the series converges

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Which of the following series converge?

A. $\sum 1/(n \log n)$
B. $\sum 1/(n \log^{2} n)$ ✓ Correct
C. $\sum (-1)^{n}/\sqrt{n}$ ✓ Correct
D. $\sum$ $1/n^{1 + 1/n}$

Solution

(a) diverges by condensation $(\sum1/(k \log 2))$ .

(b) converges by condensation $(\sum1/(k^{2} \log^{2} 2))$ .

(d) $n^{1/n}$ → 1 so terms ~ 1/n — diverges.