Real Analysis

Divergent sequences

Divergent sequences are sequences of real numbers that do not converge to any finite limit, either growing without bound, oscillating infinitely, or tending to infinity or negative infinity.

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

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Definition and statement

Definition and Characterization of Divergence to Infinity ($\pm \infty$)

A sequence $\langle x_n \rangle$ diverges to positive infinity, written as $\lim_{n \to \infty} x_n = +\infty$ (or $x_n \to \infty$), if for every real number $M > 0$, however large, there exists a positive integer $N = N(M)$ such that $x_n > M$ for all $n \ge N$.

📐 Geometric Interpretation

Geometrically, as you move further along the sequence index $n$ on the horizontal axis, the terms $x_n$ eventually rise above any arbitrarily chosen high horizontal barrier $y = M$ and never fall back below it.

⚙️ Algebraic & Structural Meaning

Provides the formal arithmetic foundation for manipulating limits involving infinity, reciprocal relations ($x_n \to \infty \iff \frac{1}{x_n} \to 0$ for $x_n > 0$), and growth rate comparisons among polynomials, exponentials, and factorials.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Positivity of Terms for Reciprocal Limit	`Domain of sequence terms $x_n > 0$ for all $n \ge N_0$`	Strictly Mandatory / Necessary	If terms alternate in sign while tending in magnitude to infinity, $\frac{1}{x_n}$ oscillates across zero or is undefined rather than converging to $0$.
Monotonicity for Monotone Convergence Theorem (Divergence Form)	`Monotonically increasing sequence ($x_{n+1} \ge x_n$)`	Essential for unbounded deduction	If monotonicity fails, an unbounded sequence may oscillate wildly (e.g., $x_n = n \sin(n\pi/2)$) without diverging to $+\infty$ or $-\infty$.
Archimedean Property of Real Numbers	`Underlying field $\mathbb{R}$`	Fundamental axiom	Without the Archimedean property, bounds $M$ could exceed all natural numbers, making divergence to infinity impossible to formalize.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Polynomial Growth Dominance

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

Rule: For rational sequences, compare the highest degree of numerator ($k$) and denominator ($j$) instantly by inspecting leading terms.

Exponential vs. Polynomial Divergence Rate

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

Rule: Exponential growth of base $a > 1$ strictly outpaces any polynomial growth $n^k$; hence $a^n / n^k \to +\infty$.

Application: Frequently asked in KVS and NVS PGT assertions regarding the relative speed of divergent sequences.

Factorial Dominance

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

Rule: Factorial $n!$ grows faster than any exponential $a^n$, meaning $n! / a^n \to +\infty$.

Application: Standard comparative limit test appearing in DSSSB PGT Mathematics.

Stirling's Asymptotic Divergence Approximation

$$ n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n $$

Rule: Use Stirling's approximation to evaluate complex factorial ratios in divergent sequence limits.

Application: Advanced UP PGT questions involving factorial limits and infinite products.

Root Test / Ratio Test Link for Sequence Growth

$$ \lim_{n \to \infty} \left|\frac{x_{n+1}}{x_n}\right| = L > 1 \implies |x_n| \to +\infty $$

Rule: If the successive ratio exceeds $1$, the sequence diverges in magnitude.

Application: Frequently bridges sequence divergence and infinite series radius of convergence problems.

Rapid Oscillation Elimination

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

Rule: An unbounded sequence with alternating signs diverges by oscillation (neither $+\infty$ nor $-\infty$), eliminating improper divergence options.

Application: Eliminates distractor options in multiple-choice questions regarding unbounded vs. properly divergent sequences.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Unboundedness vs. Divergence to Infinity Confusion

Common Pitfall: Assuming every unbounded sequence diverges to $+\infty$ or $-\infty$.

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

Why it fails: The sequence is unbounded above and below, but it oscillates infinitely between positive and negative values without tending to $+\infty$ or $-\infty$.

🛡️ Exam Defense: Unboundedness is necessary but NOT sufficient for proper divergence to $\pm \infty$. Always check sign persistence for large $n$.

⚠️ COMMON ERROR #02

Algebraic Operations on Infinity ($\infty - \infty$)

Common Pitfall: Treating $\infty$ as a real number and evaluating indeterminate forms like $\infty - \infty$ or $0 \cdot \infty$ directly.

Counterexample: $x_n = (n+1) - n = 1 \text{ vs } y_n = (2n) - n = n

Why it fails: Expressions like $\infty - \infty$ are indeterminate; the limit depends entirely on the algebraic structure before taking limits.

🛡️ Exam Defense: Always rationalize, factor out dominant terms, or apply L'Hôpital's rule (via Stolz-Cesàro theorem) before concluding.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Proper Divergence (+∞)	∀ M > 0, ∃ N s.t. x_n > M ∀ n ≥ N	lim x_n = +∞	Terms eventually cross and stay above any horizontal line y = M	Terms grow larger than any bound
Proper Divergence (-∞)	∀ M < 0, ∃ N s.t. x_n < M ∀ n ≥ N	lim x_n = -∞	Terms eventually fall below any horizontal line y = M	Terms decrease below any negative bound
Oscillatory Divergence	Sequence is bounded or unbounded, limit does not exist (not ±∞)	lim inf x_n ≠ lim sup x_n	Terms bounce between multiple cluster points without settling	Presence of (-1)^n or trigonometric terms with unbounded growth
Stolz-Cesàro Theorem	y_n is strictly monotonic and divergent to +∞	lim (x_n - x_{n-1}) / (y_n - y_{n-1})	Discrete analogue of L'Hôpital's rule for sequence ratios	Ratio of two sequences where denominator diverges to infinity

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine the limit behavior of the sequence $x_n = \frac{n^2 + 3n + 2}{2n^2 - 1}$ as $n \to \infty$.

(A) $1/2$(B) $1$(C) $2$(D) $+\infty$

📚 Method A: Formal Textbook Derivation

Divide numerator and denominator by highest power $n^2$: $x_n = \frac{1 + 3/n + 2/n^2}{2 - 1/n^2}$. Taking limit as $n \to \infty$ yields $\frac{1 + 0 + 0}{2 - 0} = \frac{1}{2}$.

⚡ Method B: 25-Second Exam Shortcut

Ratio of leading coefficients: $a_2 = 1, b_2 = 2 \implies \frac{1}{2}$. Takes 5 seconds.

💡 Tip: Check degrees of numerator and denominator first. If equal, the answer is simply the ratio of leading coefficients.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Identify the nature of the sequence $x_n = n^{1/n}$ as $n \to \infty$.

(A) Diverges to $+\infty$(B) Converges to $1$(C) Oscillates finitely(D) Converges to $0$

📚 Method A: Formal Textbook Derivation

Let $x_n = n^{1/n}$. Write $n^{1/n} = e^{\frac{\ln n}{n}}$. Since $\lim_{n \to \infty} \frac{\ln n}{n} = 0$, by continuity of the exponential function, $\lim_{n \to \infty} n^{1/n} = e^0 = 1$.

⚡ Method B: 25-Second Exam Shortcut

Standard fundamental limit memorization: $\lim n^{1/n} = 1$.

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

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Which of the following sequences diverges to $+\infty$?

A $x_n = \frac{(-1)^n}{n}$

B $x_n = \ln(n)$

C $x_n = \sin(n)$

D $x_n = \frac{1}{n^2}$

Derivation & Analysis:

The logarithmic function $\ln(n)$ grows without bound as $n \to \infty$, whereas $\frac{(-1)^n}{n}$ and $\frac{1}{n^2}$ converge to $0$, and $\sin(n)$ oscillates between $-1$ and $1$.

MCQ 2: The sequence $x_n = \frac{n!}{2^n}$ behaves as follows:

A Converges to $0$

B Converges to $1/2$

C Diverges to $+\infty$

D Oscillates finitely

Derivation & Analysis:

Factorial growth strictly dominates exponential growth. Therefore, $\lim_{n \to \infty} \frac{n!}{2^n} = +\infty$.

MCQ 3: If a sequence $\langle x_n \rangle$ diverges to $+\infty$, then the sequence $\langle \frac{1}{x_n} \rangle$ (for $x_n \neq 0$):

A Diverges to $+\infty$

B Converges to $1$

C Converges to $0$

D Diverges to $-\infty$

Derivation & Analysis:

By definition of limits to infinity, if $x_n \to +\infty$, then for any $\epsilon > 0$, we can choose $M = 1/\epsilon$, so $x_n > 1/\epsilon \implies 0 < \frac{1}{x_n} < \epsilon$, meaning $\lim \frac{1}{x_n} = 0$.

MCQ 4: The sequence $x_n = (-1)^n n$ is:

A Convergent

B Divergent to $+\infty$

C Divergent to $-\infty$

D Unbounded and oscillatory divergent

Derivation & Analysis:

The terms alternate in sign and grow in magnitude ($1, -2, 3, -4, \dots$). The sequence is unbounded and does not tend to $+\infty$ or $-\infty$, hence it is oscillatory divergent.

MCQ 5: What is $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{2n}$?

A $e$

B $e^2$

C $1$

D $+\infty$

Derivation & Analysis:

We know $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$. Raising the expression to the power of $2$ gives $e^2$.

MCQ 6: If $x_n > 0$ for all $n$ and $\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = L > 1$, then $\lim_{n \to \infty} x_n$ is:

A $0$

B $1$

C $+\infty$

D Does not exist (oscillatory)

Derivation & Analysis:

By the ratio test corollary for sequences, if successive terms grow by a factor greater than $1$ in the limit, the sequence diverges to $+\infty$.

MCQ 7: Which sequence satisfies $\lim_{n \to \infty} x_n = +\infty$ by root test comparison?

A $x_n = \left(\frac{3}{2}\right)^n$

B $x_n = \left(\frac{2}{3}\right)^n$

C $x_n = \frac{n}{n+1}$

D $x_n = \frac{\cos n}{n}$

Derivation & Analysis:

For $x_n = (\frac{3}{2})^n$, the base is greater than $1$, so $(3/2)^n \to +\infty$. The root test gives $\lim (x_n)^{1/n} = \frac{3}{2} > 1$, confirming divergence to $+\infty$.

MCQ 8: Assertion (A): Every bounded sequence is divergent. Reason (R): Boundedness precludes a sequence from tending to infinity.

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 but (R) is true

Derivation & Analysis:

Assertion (A) is false because bounded sequences can converge (e.g., $1/n$). Reason (R) is true because a sequence tending to $+\infty$ must be unbounded above.

MCQ 9: Evaluate $\lim_{n \to \infty} \frac{n^3 - 2n + 1}{3n^3 + 5n^2}$:

A $1/3$

B $0$

C $+\infty$

D $3$

Derivation & Analysis:

Both numerator and denominator are polynomials of degree $3$. The limit is the ratio of their leading coefficients: $\frac{1}{3}$.

Q10

MCQ 10: Stolz-Cesàro Theorem is applicable to evaluate the limit of sequence ratios when the denominator sequence is:

A Bounded and convergent

B Strictly monotonic and divergent to $+\infty$

C Oscillating between finite bounds

D Converging to $0$

Derivation & Analysis:

Stolz-Cesàro theorem (the discrete analogue of L'Hôpital's rule) requires the denominator sequence to be strictly monotonic and divergent to $+\infty$.

Common questions

No. An unbounded sequence can oscillate infinitely without settling on positive or negative infinity, such as $x_n = (-1)^n n$ or $x_n = n \cos(n\pi)$. Proper divergence to $\pm\infty$ requires permanent sign persistence and growth beyond any bound.

Treating indeterminate forms like $\infty - \infty$ or $0 \cdot \infty$ algebraically without proper factorization or rationalization. Always reduce expressions to standard convergent or divergent building blocks.

Master algebraic growth hierarchies ($n! \gg a^n \gg n^k \gg \ln n$) and use leading-term extraction or standard limit shortcuts to solve polynomial and exponential ratios.