Real Analysis

Properties of convergent sequences

Q: FAQ 1: Are all bounded sequences convergent?

No. Boundedness is a necessary condition for convergence, but not sufficient. For example, $a_n = (-1)^n$ is bounded between $-1$ and $1$ but does not converge because it oscillates.

Q: FAQ 2: What is the difference between a convergent sequence and a Cauchy sequence in $\mathbb{R}$?

In the real number system $\mathbb{R}$, every convergent sequence is Cauchy, and conversely, every Cauchy sequence is convergent due to the completeness property of $\mathbb{R}$.

Q: FAQ 3: How can recursive sequence limits be found quickly in exams?

Instead of proving convergence rigorously, assume the limit $l$ exists, substitute $l$ for $a_{n+1}$ and $a_n$ in the recurrence relation, and solve the resulting algebraic equation.

The properties of convergent sequences establish the fundamental algebraic, order, and topological behavior of limits in the real number system $\mathbb{R}$.

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

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Definition and statement

Algebra of Convergent Sequences (Limit Theorems)

Let $\langle a_n \rangle$ and $\langle b_n \rangle$ be sequences of real numbers such that $\lim_{n \to \infty} a_n = l$ and $\lim_{n \to \infty} b_n = m$. Then: (i) $\lim_{n \to \infty} (a_n \pm b_n) = l \pm m$, (ii) $\lim_{n \to \infty} (c a_n) = c l$ for any constant $c$, (iii) $\lim_{n \to \infty} (a_n b_n) = l m$, and (iv) $\lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = \frac{l}{m}$, provided $m \neq 0$ and $b_n \neq 0$ for all $n$.

📐 Geometric Interpretation

Geometrically, as the index $n$ advances toward infinity, the terms of the combined sequence cluster indefinitely close to a single point determined by the arithmetic combination of their individual limiting targets $l$ and $m$.

⚙️ Algebraic & Structural Meaning

These properties prove that the set of all convergent sequences forms an algebra over $\mathbb{R}$ under termwise operations, preserving limits through continuous algebraic transformations.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Existence of Individual Limits	`Entire sequence domain $n \in \mathbb{N}$`	Strictly Mandatory / Necessary	If either $\langle a_n \rangle$ or $\langle b_n \rangle$ diverges, algebraic sum/product rules fail entirely (e.g., oscillating sequences).
Non-zero Denominator Limit	`Denominator sequence $b_n \neq 0, m \neq 0$`	Strictly Mandatory	Division by zero or approaching zero results in undefined expressions or infinite limits.
Order Preservation (Inequality)	`Termwise inequality $a_n \le b_n$`	Essential for conclusion	Strict inequalities do not pass to strict inequalities at the limit; e.g., $1/n > 0$ yields limit $0 \ge 0$ (equality is permitted).

Useful results and shortcuts

Use these identities and results when their conditions apply:

Rational Function Limit Ratio

$$ \lim_{n \to \infty} \frac{a_k n^k + a_{k-1}n^{k-1} + \dots + a_0}{b_j n^j + b_{j-1}n^{j-1} + \dots + b_0} = \begin{cases} \frac{a_k}{b_j} & \text{if } k = j \\ 0 & \text{if } k < j \\ \pm\infty & \text{if } k > j \end{cases} $$

Rule: Compare highest degrees of numerator and denominator instantly to write down the limit without $\frac{1}{n}$ expansion.

Squeeze (Sandwich) Theorem for Roots

$$ \lim_{n \to \infty} (n^k + \dots + n^m)^{1/n} = 1 $$

Rule: Any algebraic polynomial or fixed sum raised to the power $1/n$ converges to $1$ as $n \to \infty$.

Exponential vs. Factorial Growth Rule

$$ \lim_{n \to \infty} \frac{a^n}{n!} = 0 \quad (\forall a \in \mathbb{R}) $$

Rule: Factorial growth strictly dominates exponential growth, which strictly dominates polynomial growth: $n^k \ll a^n \ll n! \ll n^n$.

Application: Frequently asked in comparative limit ranking questions.

Stolz-Cesaro Theorem (L'Hopital for Sequences)

$$ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{a_n - a_{n-1}}{b_n - b_{n-1}} $$

Rule: Use difference ratio when encountering indeterminate forms $\frac{\infty}{\infty}$ with strictly monotonic divergent denominators.

Application: Used for solving complex summation limits in high-level PGT papers.

Bounded Monotone Convergence

$$ \text{Monotone Increasing} + \text{Bounded Above} \implies \text{Converges to } \sup(a_n) $$

Rule: If recurrence relation $a_{n+1} = f(a_n)$ is monotone and bounded, equate $a_{n+1} = a_n = l$ and solve algebraic equation $l = f(l)$.

Rapid Elimination Trick for Oscillatory Terms

$$ \lim_{n \to \infty} \frac{\sin(n\theta)}{n} = 0 $$

Rule: Bounded oscillating factor divided by a term growing to infinity instantly evaluates to zero by Squeeze Theorem.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Converse of Boundedness

Common Pitfall: Candidates assume that every bounded sequence is convergent.

Counterexample: a_n = (-1)^n

Why it fails: The sequence is bounded ($|a_n| \le 1$) but oscillates infinitely between $-1$ and $1$, hence does not converge.

🛡️ Exam Defense: Remember: Convergent $\implies$ Bounded, but Bounded $\not\implies$ Convergent (need monotonicity or Cauchy property).

⚠️ COMMON ERROR #02

Strict Inequality Limit Fallacy

Common Pitfall: Assuming that if $a_n < b_n$ for all $n$, then $\lim a_n < \lim b_n$ strictly.

Counterexample: a_n = 0, \quad b_n = \frac{1}{n}

Why it fails: Here $0 < \frac{1}{n}$ for all $n$, but both limits are equal to $0$ ($0 \le 0$).

🛡️ Exam Defense: Strict inequalities weaken to non-strict inequalities upon taking limits: $a_n < b_n \implies \lim a_n \le \lim b_n$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Convergence	$\forall \epsilon > 0, \exists N$ s.t. $\|a_n - l\| < \epsilon$	$\|a_n - l\| \to 0$ as $n \to \infty$	Terms eventually stay inside any neighborhood of $l$	Find limit or check convergence
Boundedness	$\exists M > 0$ s.t. $\|a_n\| \le M$	$-M \le a_n \le M$	All terms confined between two horizontal lines	Test if sequence blows up to $\infty$
Monotone Convergence	Monotone + Bounded	Limit equals supremum or infimum	Sequence moves in one direction and hits a ceiling/floor	Recursive relation $a_{n+1} = f(a_n)$
Cauchy Criterion	$\|a_n - a_m\| < \epsilon$ for $n, m \ge N$	Sequence converges in $\mathbb{R}$ without knowing limit	Terms get arbitrarily close to each other	Abstract metric / completeness tests

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Evaluate the limit of the sequence $\langle a_n \rangle$ given by $a_n = \left(1 + \frac{1}{n}\right)^n$.

(A) $0$(B) $1$(C) $e$(D) $\infty$

📚 Method A: Formal Textbook Derivation

Step 1: Recognize standard limit definition of Euler's number $e$. Step 2: Apply logarithmic limit transformation $\lim_{n \to \infty} n \ln(1 + 1/n) = 1$. Step 3: Exponentiate to get $e^1 = e$.

⚡ Method B: 25-Second Exam Shortcut

Standard textbook result memorization yields the answer in 2 seconds.

💡 Tip: Direct standard result frequently asked in UP TGT/PGT.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Find the limit of the recursive sequence defined by $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + a_n}$ for $n \ge 1$.

(A) $1$(B) $2$(C) $\sqrt{2}$(D) $\sqrt{3}$

📚 Method A: Formal Textbook Derivation

Step 1: Prove boundedness by induction ($a_n < 2$). Step 2: Prove monotonicity ($a_{n+1} > a_n$). Step 3: Let $\lim a_n = l$ and solve $l = \sqrt{2 + l} \implies l^2 - l - 2 = 0 \implies l = 2$ (discarding negative root).

⚡ Method B: 25-Second Exam Shortcut

Set $a_{n+1} = a_n = l$ directly in the recurrence relation: $l = \sqrt{2 + l} \implies l = 2$.

💡 Tip: Always check initial condition to ensure positive root selection.

Practice questions

Choose an answer to check it and see the solution:

Every convergent sequence in $\mathbb{R}$ is:

A Monotone

B Bounded

C Unbounded

D Geometric progression

Derivation & Analysis:

By fundamental theorems of real analysis, every convergent sequence is bounded, though the converse is false.

The limit of the sequence $a_n = \frac{2n^2 + 3n}{5n^2 - 1}$ as $n \to \infty$ is:

A $0$

B $3/5$

C $2/5$

D $\infty$

Derivation & Analysis:

Degree of numerator and denominator are equal ($2$). The limit is the ratio of leading coefficients: $\frac{2}{5}$.

What is $\lim_{n \to \infty} \frac{n!}{n^n}$?

A $1$

B $\infty$

C $e$

D $0$

Derivation & Analysis:

By growth rate hierarchy, $n! \ll n^n$, hence the ratio tends to $0$ as $n \to \infty$.

If $a_n = \frac{(-1)^n}{n}$, then the sequence $\langle a_n \rangle$ is:

A Convergent to $1$

B Convergent to $0$

C Divergent

D Oscillatory finitely without limit

Derivation & Analysis:

By Squeeze Theorem, $-1/n \le \frac{(-1)^n}{n} \le 1/n$. As $n \to \infty$, both bounds approach $0$, so $\lim a_n = 0$.

Which of the following sequences is divergent?

A $\langle \frac{1}{n} \rangle$

B $\langle \left(1 + \frac{1}{n}\right)^n \rangle$

C $\langle (-1)^n \rangle$

D $\langle \frac{n}{n+1} \rangle$

Derivation & Analysis:

$\langle (-1)^n \rangle$ oscillates between $-1$ and $1$ and does not approach a unique limit, hence divergent.

If $a_n > 0$ for all $n$ and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = l < 1$, then $\lim_{n \to \infty} a_n$ is:

A $1$

B $0$

C $l$

D $\infty$

Derivation & Analysis:

D'Alembert's Ratio Test for sequences: if ratio of successive terms is less than $1$ in limit, the sequence converges to $0$.

The sequence $a_n = n^{1/n}$ converges to:

A $0$

B $1$

C $e$

D $\infty$

Derivation & Analysis:

Standard limit result: $\lim_{n \to \infty} n^{1/n} = 1$, established using logarithms or Cauchy's First Theorem on Limits.

Let $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$. The sequence $\langle a_n \rangle$ is:

A Convergent to $e$

B Divergent to $\infty$

C Oscillatory

D Convergent to $1$

Derivation & Analysis:

This is the standard sequence representation of the number $e$, which is monotone increasing and bounded above by $3$.

If $a_n \le b_n \le c_n$ for all $n \ge N$ and $\lim a_n = \lim c_n = L$, then $\lim b_n$ equals:

A $0$

B $L$

C Does not exist

D $2L$

Derivation & Analysis:

This is the direct statement of the Squeeze (Sandwich) Theorem for sequences.

Q10

The value of $\lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \dots + \frac{n}{n^2} \right)$ is:

A $0$

B $1$

C $1/2$

D $2$

Derivation & Analysis:

Sum is $\frac{n(n+1)}{2n^2} = \frac{n^2+n}{2n^2}$. Taking limit as $n \to \infty$ gives $\frac{1}{2}$.

Common questions

No. Boundedness is a necessary condition for convergence, but not sufficient. For example, $a_n = (-1)^n$ is bounded between $-1$ and $1$ but does not converge because it oscillates.

In the real number system $\mathbb{R}$, every convergent sequence is Cauchy, and conversely, every Cauchy sequence is convergent due to the completeness property of $\mathbb{R}$.

Instead of proving convergence rigorously, assume the limit $l$ exists, substitute $l$ for $a_{n+1}$ and $a_n$ in the recurrence relation, and solve the resulting algebraic equation.