Real Analysis

Limit of sequences

Q: FAQ 1: Is every bounded sequence convergent?

No. Boundedness is necessary but not sufficient. For example, $x_n = (-1)^n$ is bounded between $-1$ and $1$ but does not converge. It must also be monotonic to guarantee convergence.

Q: FAQ 2: What is the most common reason students lose marks on limit questions?

Applying algebraic limit laws (like sum or product rules) when individual limits do not exist or are infinite, leading to indeterminate forms.

Q: FAQ 3: How can candidates solve recursive sequence limit problems?

Skip proving monotonicity and boundedness in the exam hall; directly equate $x_{n+1}$ and $x_n$ to $l$, solve the resulting algebraic equation, and select the physically valid root matching initial terms.

The limit of a sequence is the fundamental analytical concept defining the behavior of terms $x_n$ as $n$ tends to infinity. This topic covers $\epsilon-N$ definitions, algebra of limits, and convergence criteria.

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

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Definition and statement

Definition of Convergence of a Sequence

A sequence $\langle x_n \rangle$ of real numbers is said to converge to a real number $l$ if for every $\epsilon > 0$, however small, there exists a positive integer $N = N(\epsilon)$ such that $|x_n - l| < \epsilon$ for all $n \ge N$. Symbolically, $\lim_{n \to \infty} x_n = l$ or $x_n \to l as n \to \infty$.

📐 Geometric Interpretation

Geometrically, for any arbitrarily small strip $(l - \epsilon, l + \epsilon)$ around the limit $l$, all terms of the sequence from some point $N$ onwards lie strictly inside this neighborhood.

⚙️ Algebraic & Structural Meaning

It translates infinite asymptotic behavior into finite arithmetic bounds using inequalities, enabling rigorous proofs of continuity, series convergence, and bounds.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Archimedean Property of $\mathbb{R}$	`Domain of indices $n \in \mathbb{N}$`	Strictly Mandatory / Necessary	Without it, bounds like $1/n < \epsilon$ cannot be guaranteed to yield a natural number index $N$.
Boundedness of Convergent Sequences	`Global range $\{x_n : n \in \mathbb{N}\- \subset \mathbb{R}$`	Strictly Necessary (Necessary condition, not sufficient)	Unbounded sequences (e.g., $x_n = n$) diverge to infinity and cannot possess a finite limit.
Uniqueness of Limit	`Limit point uniqueness`	Essential for well-definedness	If violated, limits could oscillate or be non-unique, destroying algebraic limit laws.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Algebraic Growth Rate Hierarchy

$$ \lim_{n \to \infty} \frac{(\ln n)^k}{n^p} = 0 \quad (p > 0, k > 0), \quad \lim_{n \to \infty} \frac{n^k}{a^n} = 0 \quad (a > 1) $$

Rule: Logarithms grow slower than polynomials, polynomials grow slower than exponentials, and exponentials grow slower than factorials ($n!$).

Application: Directly used in UP PGT/TGT to find limits of ratios containing mixed growth terms instantly without L'Hôpital's rule extensions.

Cauchy's First Theorem on Limits (Cesàro Means)

$$ \text{If } \lim_{n \to \infty} x_n = l, \text{ then } \lim_{n \to \infty} \frac{x_1 + x_2 + \dots + x_n}{n} = l $$

Rule: The limit of the arithmetic mean of sequence terms equals the limit of the terms themselves.

Application: Frequently asked in DSSSB and KVS PGT for complex summation limits.

Cauchy's Second Theorem on Limits

$$ \text{If } x_n > 0 \text{ and } \lim_{n \to \infty} \frac{x_{n+1}}{x_n} = l, \text{ then } \lim_{n \to \infty} (x_n)^{1/n} = l $$

Rule: Convert $n$-th root limits into ratio limits to simplify radical evaluations.

Application: Standard pattern in UP PGT for evaluating limits like $\lim_{n \to \infty} (n!)^{1/n} / n$.

Squeeze (Sandwich) Theorem for Sequences

$$ \text{If } a_n \le x_n \le b_n \text{ and } \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = l, \text{ then } \lim_{n \to \infty} x_n = l $$

Rule: Bound complicated expressions by two simple sequences sharing the same limit.

Monotone Convergence Theorem (MCT) Quick Test

$$ \text{Bounded + Monotone} \implies \text{Convergent} $$

Rule: For recursively defined sequences $x_{n+1} = f(x_n)$, check monotonicity by $x_{n+1} - x_n$ or $x_{n+1}/x_n$, then solve $l = f(l)$ for the limit.

Application: Guaranteed 1 question in UP PGT/TGT regarding recursive square root or fractional sequences.

Rapid Elimination via Subsequences

$$ \text{If } \lim_{n \to \infty} x_n = l, \text{ every subsequence } x_{n_k} \text{ also converges to } l. $$

Rule: If even two distinct subsequences converge to different limits, the main sequence diverges.

Application: Used to eliminate oscillating options like $(-1)^n$ or $\sin(n\pi/2)$ instantly.

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: Assuming that every bounded sequence is convergent.

Counterexample: $x_n = (-1)^n$

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

🛡️ Exam Defense: Remember: Boundedness is necessary for convergence, but NOT sufficient. Monotonicity must be present.

⚠️ COMMON ERROR #02

Algebraic Operations on Divergent Sequences

Common Pitfall: Applying $\lim(a_n + b_n) = \lim a_n + \lim b_n$ when individual limits do not exist.

Counterexample: Let $a_n = n$ and $b_n = -n$.

Why it fails: $a_n + b_n = 0$, whose limit is $0$, but $\lim a_n$ and $\lim b_n$ do not exist finitely.

🛡️ Exam Defense: Limit laws apply ONLY when individual limits are finite and existent.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Monotone Convergence Theorem	Sequence is monotonic and bounded	Limit exists and equals supremum/infimum	Staircase graph approaches a horizontal asymptote	Recursive formulas $x_{n+1} = f(x_n)
Sandwich Theorem	$a_n \le x_n \le b_n$ and $\lim a_n = \lim b_n = l$	$\lim x_n = l$	Curve squeezed between two bounding curves	Sums of $n$ terms divided by $n$ or root terms
Cauchy's First Theorem	$\lim x_n = l$	$\lim \frac{1}{n}\sum x_i = l$	Average value stabilizes to the limit	Arithmetic mean of sequence terms
Cauchy's Criterion	Sequence is in metric space $\mathbb{R}$	$\|x_m - x_n\| < \epsilon$ for $m,n \ge N$	Terms cluster infinitely close together without needing the limit value	Proving convergence without knowing the limit

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Evaluate 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

📚 Method A: Formal Textbook Derivation

Step 1: Prove boundedness by induction ($x_n < 2$). Step 2: Prove monotonicity ($x_{n+1} > x_n$). Step 3: Let $\lim x_n = l$. Step 4: Solve $l = \sqrt{2 + l} \implies l^2 - l - 2 = 0 \implies l = 2$ (since $l > 0$).

⚡ Method B: 25-Second Exam Shortcut

Directly set $x = \sqrt{2+x}$ and solve $x^2 - x - 2 = 0 \implies x = 2$.

💡 Tip: For recursive root problems of type $\sqrt{a + \sqrt{a + \dots}}$, the positive root of $x^2 - x - a = 0$ is always the answer if $x_1 > 0$.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Find $\lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{n+n} \right)$.

(A) 0(B) 1(C) \log 2(D) e

📚 Method A: Formal Textbook Derivation

Rewrite as $\sum_{r=1}^n \frac{1}{n} \frac{1}{1 + r/n}$. Convert to definite integral $\int_0^1 \frac{dx}{1+x} = [\ln(1+x)]_0^1 = \ln 2$.

⚡ Method B: 25-Second Exam Shortcut

Recognize Riemann sum limit: $\lim \frac{1}{n}\sum f(r/n) = \int_0^1 f(x)dx$.

💡 Tip: Sum of $n$ terms with denominator $n+r$ always converts to $\int_0^1 \frac{1}{1+x} dx = \ln 2$.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The sequence $x_n = \frac{n}{n+1}$ is:

A Monotonic increasing and bounded

B Monotonic decreasing and unbounded

C Not monotonic but bounded

D Divergent

Derivation & Analysis:

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

MCQ 2: What is $\lim_{n \to \infty} \frac{n!}{n^n}$?

A 1

B e

C 0

D $\infty$

Derivation & Analysis:

By the growth hierarchy, $n!$ grows much slower than $n^n$, hence the ratio tends to $0$. Rigorously, use sandwich theorem between $0$ and $\frac{1}{n} \cdot \frac{2}{n} \dots$.

MCQ 3: The limit of the sequence $x_n = (-1)^n \frac{1}{n}$ is:

A 1

B -1

C Does not exist

D 0

Derivation & Analysis:

By sandwich theorem, $-\frac{1}{n} \le (-1)^n \frac{1}{n} \le \frac{1}{n}$. Since limits of bounds are both $0$, the limit is $0$.

MCQ 4: If $x_n = \left(1 + \frac{1}{n}\right)^n$, then $\lim_{n \to \infty} x_n$ is:

A 1

B e

C 0

D $\infty$

Derivation & Analysis:

This is the standard definition of Euler's number $e$ as a sequence limit, converging monotonically to $e$.

MCQ 5: Which of the following sequences is convergent?

A $x_n = n^{1/n}$

B $x_n = (-1)^n n$

C $x_n = \ln n$

D $x_n = 2^n$

Derivation & Analysis:

$n^{1/n} \to 1$ as $n \to \infty$ (a standard standard limit). Options B, C, D diverge to infinity or oscillate.

MCQ 6: The sequence $x_n = \sin\left(\frac{n\pi}{2}\right)$ is:

A Convergent to 1

B Convergent to 0

C Oscillatory (divergent)

D Convergent to -1

Derivation & Analysis:

The terms are $1, 0, -1, 0, 1, 0, -1, \dots$, which oscillate between multiple values without settling, hence divergent.

MCQ 7: If $\lim_{n \to \infty} x_n = 5$, then $\lim_{n \to \infty} \frac{2x_n + 3}{x_n - 1}$ equals:

A 13/4

B 5

C 2

D 1

Derivation & Analysis:

Substitute limit directly: $\frac{2(5) + 3}{5 - 1} = \frac{13}{4}$.

MCQ 8: Cauchy's criterion for convergence of a sequence states that $x_n$ converges if and only if for every $\epsilon > 0$, there exists $N$ such that:

A $|x_n - l| < \epsilon$

B $|x_{n+p} - x_n| < \epsilon \quad \forall n \ge N, p \ge 1$

C $x_n < \epsilon$

D $|x_n| < N$

Derivation & Analysis:

Cauchy's general principle of convergence states $|x_m - x_n| < \epsilon$ for all $m, n \ge N$ without requiring knowledge of the limit $l$.

MCQ 9: Evaluate $\lim_{n \to \infty} \frac{1^2 + 2^2 + \dots + n^2}{n^3}$:

A 1

B 1/2

C 1/3

D 0

Derivation & Analysis:

Sum of squares is $\frac{n(n+1)(2n+1)}{6}$. Dividing by $n^3$ gives $\frac{2n^3 + 3n^2 + n}{6n^3} \to \frac{2}{6} = \frac{1}{3}$.

Q10

MCQ 10: If $0 < y < x$, and $x_n = \frac{x^n + y^n}{x^n - y^n}$, then $\lim_{n \to \infty} x_n$ is:

A 0

B 1

C -1

D $\infty$

Derivation & Analysis:

Divide numerator and denominator by $x^n$: $\lim \frac{1 + (y/x)^n}{1 - (y/x)^n}$. Since $y < x$, $(y/x)^n \to 0$, giving $\frac{1+0}{1-0} = 1$.

Common questions

No. Boundedness is necessary but not sufficient. For example, $x_n = (-1)^n$ is bounded between $-1$ and $1$ but does not converge. It must also be monotonic to guarantee convergence.

Applying algebraic limit laws (like sum or product rules) when individual limits do not exist or are infinite, leading to indeterminate forms.

Skip proving monotonicity and boundedness in the exam hall; directly equate $x_{n+1}$ and $x_n$ to $l$, solve the resulting algebraic equation, and select the physically valid root matching initial terms.