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Real Analysis

Monotonic sequences

A monotonic sequence is one that moves exclusively in a single direction—either non-decreasing or non-increasing—forming the structural bedrock of real analysis and convergence theory. This topic covers monotonic sequences and their boundedness.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade
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Definition and statement

Monotone Convergence Theorem (MCT) for Sequences
Every bounded monotonic sequence of real numbers is convergent. Specifically, if $\langle x_n \rangle$ is monotonically increasing and bounded above, then $\lim_{n \to \infty} x_n = \sup \{x_n : n \in \mathbb{N}\}$. If $\langle x_n \rangle$ is monotonically decreasing and bounded below, then $\lim_{n \to \infty} x_n = \inf \{x_n : n \in \mathbb{N}\}.

📐 Geometric Interpretation

Geometrically, the terms of the sequence either constantly march upwards or downwards on the real number line while being trapped by a finite barrier. Because they cannot jump past their supremum or infimum and never reverse direction, they must accumulate and settle precisely at their least upper bound or greatest lower bound.

⚙️ Algebraic & Structural Meaning

It bridges the algebraic order structure of the real numbers $\mathbb{R}$ with its topological completeness property. It provides a way to establish convergence without ever needing to explicitly compute or guess the limiting value beforehand.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition Domain / Interval Necessity What Fails If Condition Dropped
Monotonicity (Order preservation) Domain $\mathbb{N}$ (Discrete ordered index) Strictly Mandatory / Necessary If monotonicity fails (e.g., oscillating sequences like $(-1)^n$), boundedness alone is completely insufficient to guarantee convergence.
Boundedness (Upper or Lower bound) Subset of $\mathbb{R}$ Strictly Mandatory An unbounded monotonic sequence (like $x_n = n$) diverges to infinity, violating convergence despite perfect monotonicity.
Completeness Axiom of $\mathbb{R}$ Ordered Field Axiom Underlying Foundation In incomplete ordered fields like $\mathbb{Q}$, bounded monotonic sequences (e.g., decimal approximations of $\sqrt{2}$) fail to converge within the set.

Useful results and shortcuts

Use these identities and results when their conditions apply:

01

Ratio Test for Monotonicity

$$ \frac{x_{n+1}}{x_n} \ge 1 \implies \text{Increasing (for } x_n > 0\text{)}, \quad \frac{x_{n+1}}{x_n} \le 1 \implies \text{Decreasing (for } x_n > 0\text{)} $$

Rule: For positive term sequences, compute $x_{n+1}/x_n$ directly. If the ratio $\ge 1$ for all $n$, the sequence is monotonically increasing. Avoid tedious subtraction for radical or factorial terms.

Application: Frequently asked in UP PGT for recursive sequences like $x_{n+1} = \sqrt{c + x_n}$.
02

Boundedness via Induction & First Two Terms

$$ \text{If } x_1 < x_2 \text{ and } x_{n+1} = f(x_n) \text{ with } f \text{ increasing, then } \langle x_n \rangle \text{ is increasing.} $$

Rule: Check $x_1$ and $x_2$. If $x_2 \ge x_1$ and $f$ is an increasing function, the sequence is monotonically increasing. The upper bound is usually the positive root of the characteristic equation $x = f(x)$.

Application: Standard pattern in DSSSB and KVS PGT for determining limits of recursive sequences.
03

Limit of Monotone Recursive Sequences ($x_{n+1} = \sqrt{k + x_n}$)

$$ L = \frac{1 + \sqrt{1 + 4k}}{2} $$

Rule: For sequences defined by $x_1 > 0$ and $x_{n+1} = \sqrt{k + x_n}$, the limit is always the positive root of $L^2 - L - k = 0$.

04

Exponential vs. Factorial Growth Monotonicity

$$ \frac{x_{n+1}}{x_n} = \frac{a^{n+1}/(n+1)!}{a^n/n!} = \frac{a}{n+1} \to 0 < 1 $$

Rule: For sequences involving $n!$ and $a^n$, evaluate $x_{n+1}/x_n$. For large $n$, this ratio drops below $1$, proving the sequence is eventually monotonically decreasing and converges to $0$.

05

Algebraic Difference Test

$$ x_{n+1} - x_n = g(n) $$

Rule: If $x_{n+1} - x_n \ge 0$ for all $n \ge N$, the sequence is eventually monotonically increasing. Check the sign of the difference function directly by treating $n$ as a continuous variable $x$ and finding its derivative.

Application: Used for rational polynomial sequence terms in KVS PGT.
06

Rapid Elimination via Boundedness

$$ \text{Divergent options if unbounded} $$

Rule: If a sequence is monotonically increasing and you spot a term that grows without bound or exceeds the prospective upper bound test, immediately eliminate options suggesting convergence to a finite value.

Application: Crucial for time management across 125 questions in UP PGT.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Monotonicity without Boundedness

Common Pitfall: Assuming every monotonic sequence converges.

Counterexample: $x_n = n$ (Monotonically increasing)

Why it fails: The sequence is strictly increasing, but it is not bounded above; hence it diverges to $+\infty$.

🛡️ Exam Defense: Always verify both monotonicity AND boundedness before applying the Monotone Convergence Theorem.
⚠️ COMMON ERROR #02

Boundedness Alone Implies Convergence

Common Pitfall: Assuming any bounded sequence is convergent without checking monotonicity.

Counterexample: $x_n = (-1)^n$ on $[-1, 1]$

Why it fails: It is bounded between $-1$ and $1$, but it oscillates infinitely and is not monotonic, so it does not converge.

🛡️ Exam Defense: Boundedness guarantees the existence of convergent subsequences (Bolzano-Weierstrass), but not the convergence of the sequence itself unless monotonicity is present.

Comparison

Concept / TheoremHypotheses / AssumptionsKey Formula / OutcomeGeometric MeaningExam Identification Clue
Monotone Convergence TheoremMonotonic + Bounded$\lim x_n = \sup/\inf$Trapped curve settling at boundaryRecursive or algebraic inequalities
Cauchy's General PrincipleSequence in $\mathbb{R}$$|x_m - x_n| < \varepsilon$Terms bunch infinitely close togetherGeneral limit existence without target
Bolzano-Weierstrass TheoremBounded Sequence in $\mathbb{R}$$\exists$ convergent subsequenceInfinite points clustering in a finite windowSubsequence questions or bounded bounds
Sandwich (Squeeze) Theorem$a_n \le x_n \le b_n$ and $a_n \to L, b_n \to L$$x_n \to L$Pushed to same limit between two boundsTrigonometric ratios divided by $n$

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine the limit of the sequence defined recursively by $x_1 = \sqrt{2}$ and $x_{n+1} = \sqrt{2 + x_n}$ for $n \ge 1$.

(A) $1$(B) $2$(C) $\sqrt{2}$(D) $\infty$
✓ Correct Answer: (B)
📚 Method A: Formal Textbook Derivation
Step 1: Show $x_n < 2$ by mathematical induction. Step 2: Show $x_{n+1} > x_n$ (monotonically increasing). Step 3: By MCT, limit $L$ exists. Step 4: Solve $L = \sqrt{2 + L} \implies L^2 - L - 2 = 0 \implies L = 2$ (discarding negative root).
⚡ Method B: 25-Second Exam Shortcut
Apply Shortcut 3 directly: $k = 2$, solve $L = \frac{1 + \sqrt{1 + 4(2)}}{2} = \frac{1 + 3}{2} = 2$. Takes 10 seconds.
💡 Tip: Look for nested radical recursive patterns. The limit is always the positive root of $L^2 - L - k = 0$.
CANONICAL PROBLEM 02 Dual-Method Breakdown

If $x_n = \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n}$, then the sequence $\langle x_n \rangle$ is:

(A) Monotonically increasing and divergent(B) Monotonically increasing and convergent(C) Monotonically decreasing and convergent(D) Not monotonic
✓ Correct Answer: (B)
📚 Method A: Formal Textbook Derivation
Compute $x_{n+1} - x_n = \frac{1}{2n+1} + \frac{1}{2n+2} - \frac{1}{n+1} = \frac{1}{2n+1} - \frac{1}{2n+2} > 0$. Thus it is monotonically increasing. Boundedness by $1$ is verified, confirming convergence.
⚡ Method B: 25-Second Exam Shortcut
Recognize harmonic sum differences. The difference is positive, establishing strict increase. Bounded above by $\ln 2$, so it converges.
💡 Tip: Harmonic tail sums of this form are classic UP PGT favorites.

Practice questions

Choose an answer to check it and see the solution:

Q1
MCQ 1: Every bounded monotonic sequence in $\mathbb{R}$ is:
A Divergent
B Convergent
C Oscillatory
D Periodic
Derivation & Analysis:

This is the direct statement of the Monotone Convergence Theorem (MCT).

Q2
MCQ 2: The sequence $x_n = \frac{n}{n+1}$ is:
A Monotonically decreasing
B Monotonically increasing and bounded
C Unbounded
D Oscillating
Derivation & Analysis:

$x_{n+1} - x_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{1}{(n+1)(n+2)} > 0$, so it is increasing, and $x_n < 1$, so it is bounded.

Q3
MCQ 3: If $x_1 = 1$ and $x_{n+1} = \frac{1}{3}(x_n + 1)$, then $\lim_{n \to \infty} x_n$ is equal to:
A $\frac{1}{2}$
B $1$
C $\frac{1}{3}$
D $2$
Derivation & Analysis:

Let $L = \frac{1}{3}(L + 1) \implies 3L = L + 1 \implies 2L = 1 \implies L = \frac{1}{2}$.

Q4
MCQ 4: Which of the following sequences is monotonically decreasing?
A $x_n = n^2$
B $x_n = 2^n$
C $x_n = \frac{1}{n}$
D $x_n = \ln(n)$
Derivation & Analysis:

As $n$ increases, $\frac{1}{n}$ decreases strictly, satisfying $x_{n+1} < x_n$.

Q5
MCQ 5: A monotonic sequence which is not bounded above:
A Converges to $0$
B Diverges to $+\infty$
C Converges to $1$
D Oscillates finitely
Derivation & Analysis:

By definition, an increasing sequence not bounded above tends to $+\infty$.

Q6
MCQ 6: The sequence $x_n = (-1)^n \cdot n$ is:
A Monotone and bounded
B Monotone and unbounded
C Not monotone and unbounded
D Convergent
Derivation & Analysis:

Terms alternate signs and grow in magnitude, making it non-monotonic and unbounded.

Q7
MCQ 7: If $x_n = \left(1 + \frac{1}{n}\right)^n$, then the sequence $\langle x_n \rangle$ is:
A Monotonically decreasing
B Monotonically increasing and bounded above by $3$
C Unbounded
D Oscillatory
Derivation & Analysis:

This is Euler's sequence defining $e$, which is strictly increasing and bounded above by $3$.

Q8
MCQ 8: Assertion (A): Every convergent sequence is monotonic. Reason (R): Limit exists for monotonic bounded sequences.
A Both (A) and (R) are true and (R) is the correct explanation of (A)
B Both (A) and (R) are true but (R) is not the correct explanation of (A)
C (A) is true but (R) is false
D (A) is false and (R) is true
Derivation & Analysis:

Assertion (A) is false because convergent sequences (e.g., $\frac{(-1)^n}{n}$) need not be monotonic. Reason (R) is true.

Q9
MCQ 9: The limit of the sequence $x_n = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dots}}}$ is:
A $3$
B $4$
C $6$
D $12$
Derivation & Analysis:

Using Shortcut 3 with $k=12$: $L^2 - L - 12 = 0 \implies (L-4)(L+3) = 0 \implies L = 4$.

Q10
MCQ 10: If a sequence $\langle x_n \rangle$ is bounded below and monotonically decreasing, then its limit is:
A Supremum
B Infimum
C Zero
D First term $x_1$
Derivation & Analysis:

By the Monotone Convergence Theorem, a monotonically decreasing sequence bounded below converges to its greatest lower bound (infimum).

Common questions

Yes, but only if the sequence is constant ($x_n = c$ for all $n$). In that case, $x_{n+1} \ge x_n$ and $x_{n+1} \le x_n$ both hold simultaneously.

Failing to establish boundedness before applying algebraic limit formulas to recursive relations, or assuming non-negativity automatically guarantees monotonicity without checking $x_{n+1} - x_n$ or $x_{n+1}/x_n$.

Memorize standard radical recurrence templates like $x_{n+1} = \sqrt{k + x_n}$ where the answer is the positive root of $L^2 - L - k = 0$, bypassing cumbersome induction proofs.

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