Sequences of Real Numbers

Sequences of Real Numbers - Part 1: Introduction

Chapter Overview

Sequences of real numbers are fundamental in real analysis, forming the basis for understanding continuity, differentiation, and integration. This introductory part covers the definition of sequences, their basic properties like boundedness and monotonicity, and the crucial concept of convergence and limits. We will explore various types of sequences (convergent, divergent, oscillatory) and essential theorems like the Algebra of Limits and the Monotone Convergence Theorem, which are vital for determining the behavior of sequences. Understanding these foundational concepts is key to tackling more advanced topics and solving problems in the CUET PG exam.

Key Concepts

Definition of a Sequence

A sequence of real numbers is a function $f: \mathbb{N} \to \mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. The image of $n$ under $f$ is denoted by $a_n$ (or $x_n$, $s_n$, etc.) and is called the $n$-th term of the sequence. The sequence itself is typically denoted by $\{a_n\}$, $(a_n)$, or $a_1, a_2, a_3, \dots$.

* Example:
* $\{n\}$ represents the sequence $1, 2, 3, \dots$
* $\{\frac{1}{n}\}$ represents the sequence $1, \frac{1}{2}, \frac{1}{3}, \dots$
* $\{(-1)^n\}$ represents the sequence $-1, 1, -1, 1, \dots$

Bounded Sequences

A sequence $\{a_n\}$ is:
* Bounded Above: If there exists a real number $M$ such that $a_n \le M$ for all $n \in \mathbb{N}$. $M$ is called an upper bound.
* Bounded Below: If there exists a real number $m$ such that $a_n \ge m$ for all $n \in \mathbb{N}$. $m$ is called a lower bound.
* Bounded: If it is both bounded above and bounded below. This means there exists a real number $K > 0$ such that $|a_n| \le K$ for all $n \in \mathbb{N}$.

* Example:
* $\{\frac{1}{n}\}$ is bounded (e.g., $0 \le \frac{1}{n} \le 1$).
* $\{n\}$ is bounded below (e.g., $n \ge 1$) but not bounded above.
* $\{(-1)^n\}$ is bounded (e.g., $-1 \le (-1)^n \le 1$).

Monotonic Sequences

A sequence $\{a_n\}$ is:
* Monotonically Increasing (Non-decreasing): If $a_{n+1} \ge a_n$ for all $n \in \mathbb{N}$.
* Strictly Increasing: If $a_{n+1} > a_n$ for all $n \in \mathbb{N}$.
* Monotonically Decreasing (Non-increasing): If $a_{n+1} \le a_n$ for all $n \in \mathbb{N}$.
* Strictly Decreasing: If $a_{n+1} < a_n$ for all $n \in \mathbb{N}$.
* Monotonic: If it is either monotonically increasing or monotonically decreasing.

* Example:
* $\{n\}$ is strictly increasing.
* $\{\frac{1}{n}\}$ is strictly decreasing.
* $\{(-1)^n\}$ is not monotonic.
* $\{1, 1, 2, 2, 3, 3, \dots\}$ is monotonically increasing.

Limit of a Sequence (Convergence)

A sequence $\{a_n\}$ is said to converge to a real number $L$ (called the limit of the sequence) if for every $\epsilon > 0$, there exists a natural number $N$ (which may depend on $\epsilon$) such that for all $n > N$, $|a_n - L| < \epsilon$.
We write this as $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$.

* Formal Definition:

* Example:
* $\lim_{n \to \infty} \frac{1}{n} = 0$.
* $\lim_{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1$.

Convergent, Divergent, and Oscillatory Sequences

* Convergent Sequence: A sequence that has a finite limit $L$.
* Divergent Sequence: A sequence that is not convergent.
* Diverges to $\infty$: If for every $M > 0$, there exists $N \in \mathbb{N}$ such that $a_n > M$ for all $n > N$. We write $\lim_{n \to \infty} a_n = \infty$.
* Diverges to $-\infty$: If for every $M < 0$, there exists $N \in \mathbb{N}$ such that $a_n < M$ for all $n > N$. We write $\lim_{n \to \infty} a_n = -\infty$.
* Oscillatory Sequence: A sequence that is neither convergent nor divergent to $\pm \infty$.
* Finitely Oscillatory: An oscillatory sequence that is bounded (e.g., $\{(-1)^n\}$).
* Infinitely Oscillatory: An oscillatory sequence that is unbounded (e.g., $\{(-1)^n n\}$).

* Example:
* $\{\frac{1}{n}\}$ is convergent to $0$.
* $\{n\}$ is divergent to $\infty$.
* $\{(-1)^n\}$ is finitely oscillatory.
* $\{(-1)^n n\}$ is infinitely oscillatory.

Uniqueness of Limit

If a sequence converges, its limit is unique. A sequence cannot converge to two different limits.

Algebra of Limits

If $\{a_n\}$ and $\{b_n\}$ are two convergent sequences such that $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = M$, then:
* Sum/Difference Rule:

* Product Rule:

* Scalar Multiple Rule: For any real number $c$,

* Quotient Rule: If $M \ne 0$ and $b_n \ne 0$ for all $n$,

* Power Rule: If $L > 0$ and $p/q$ is a rational number,

Standard Limits

These limits are frequently used and should be memorized:
* For $p > 0$:

* For $|x| < 1$:

* For $x > 1$:

* For $x \le -1$: $\lim_{n \to \infty} x^n$ does not exist.
* For $a > 0$:

*
* For any real number $x$:

* A special case of the above ($x=1$):

Squeeze Theorem (Sandwich Theorem)

If $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ are sequences such that $a_n \le b_n \le c_n$ for all $n \ge N_0$ (for some $N_0 \in \mathbb{N}$), and if $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$.

* Example: To find $\lim_{n \to \infty} \left[ \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right]$:
Let $b_n = \sum_{k=1}^n \frac{1}{\sqrt{n^2+k}}$.
We know that for $1 \le k \le n$:

Summing $n$ terms:


Now, evaluate the limits of the bounding sequences:


By the Squeeze Theorem, $\lim_{n \to \infty} b_n = 1$.

Monotone Convergence Theorem (MCT)

Every bounded monotonic sequence is convergent.
* Specifically:
* A monotonically increasing sequence that is bounded above converges to its supremum.
* A monotonically decreasing sequence that is bounded below converges to its infimum.

Important Note: A monotonic sequence must* be bounded to converge. For example, $\{n\}$ is monotonic but not bounded above, so it diverges. $\left\{(-1)^n \cdot \frac{n+1}{n}\right\}$ is bounded (between $-2$ and $2$) but not monotonic, and it does not converge.

Examples for Limit Calculation

Algebraic Manipulation (Rationalization):

Find the value of $\lim_{n \to \infty} (\sqrt{4n^2 + n} - 2n)$.

Product Limit (Telescoping Product):

If $S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right)$, then $S$ is equal to:
Let $P_n = \prod_{k=2}^n \left(1-\frac{1}{k^2}\right)$.
The general term is $1-\frac{1}{k^2} = \frac{k^2-1}{k^2} = \frac{(k-1)(k+1)}{k \cdot k}$.

This is a telescoping product. Many terms cancel out:


Therefore,

Product Limit (Cauchy's Second Theorem on Limits for Products):

The value of $\lim_{n \to \infty} \left[\frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n\right]^{1/n}$ is:
Let $P_n = \frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n$. We need to find $\lim_{n \to \infty} (P_n)^{1/n}$.
This can be solved using Cauchy's Second Theorem on Limits for Products, which states that if $\lim_{n \to \infty} x_n = L$, then $\lim_{n \to \infty} (x_1 x_2 \dots x_n)^{1/n} = L$.
However, the given expression is $\left(\prod_{k=1}^n b_k\right)^{1/n}$ where $b_k = \left(\frac{k+1}{k}\right)^k$.
So we need to find $\lim_{k \to \infty} b_k$.

By Cauchy's Second Theorem on Limits for Products,

Important Points/Tips for Exam Preparation

* Master Definitions: A strong understanding of the formal definitions of convergence, boundedness, and monotonicity is crucial. Many questions test conceptual clarity.
* Practice Limit Calculations: Be proficient in evaluating limits using algebraic manipulation (rationalization, dividing by the highest power of $n$), standard limits, and the Squeeze Theorem.
* Memorize Standard Limits: Knowing the common limits (e.g., $\lim_{n \to \infty} n^{1/n}=1$, $\lim_{n \to \infty} (1+x/n)^n=e^x$) saves time.
Understand Monotone Convergence Theorem: This theorem is a cornerstone. Remember that a sequence must be both bounded and* monotonic to guarantee convergence. A common trap is to assume monotonicity alone implies convergence.
* Identify Sequence Types: Be able to classify sequences as convergent, divergent (to $\pm \infty$), or oscillatory (finitely or infinitely).
* Algebra of Limits: Apply these rules correctly. Be careful with the quotient rule when the denominator's limit is zero.
* PYQ Analysis: The provided PYQs highlight the importance of:
* Direct limit calculations (rationalization, standard forms).
* Squeeze Theorem applications.
* Conceptual understanding of convergence, monotonicity, and boundedness (Assertion-Reason type questions).
* Advanced limit techniques involving products (telescoping products, Cauchy's theorems).
* Counterexamples: Keep simple counterexamples in mind (e.g., $\{(-1)^n\}$ for a bounded but non-convergent sequence, $\{n\}$ for a monotonic but divergent sequence).
* Don't Confuse with Series: Sequences are lists of numbers; series are sums of sequences. While related, their convergence criteria are distinct.

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Chapter Overview

A sequence of real numbers is a function $f: \mathbb{N} \to \mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. The values $f(n)$ are usually denoted by $a_n$ and the sequence itself by $\{a_n\}$ or $(a_n)$. Sequences are fundamental to real analysis, forming the basis for understanding convergence, series, and continuity. This chapter covers the essential definitions, properties, and theorems related to the convergence and divergence of sequences of real numbers.

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Key Concepts

Bounded Sequences

* Bounded Above: A sequence $\{a_n\}$ is bounded above if there exists $M \in \mathbb{R}$ such that $a_n \le M$ for all $n \in \mathbb{N}$. $M$ is an upper bound.
* Bounded Below: A sequence $\{a_n\}$ is bounded below if there exists $m \in \mathbb{R}$ such that $a_n \ge m$ for all $n \in \mathbb{N}$. $m$ is a lower bound.
* Bounded Sequence: A sequence is bounded if it is both bounded above and bounded below. Equivalently, there exists $K > 0$ such that $|a_n| \le K$ for all $n \in \mathbb{N}$.
* Supremum (Least Upper Bound): The smallest of all upper bounds.
* Infimum (Greatest Lower Bound): The largest of all lower bounds.

Convergent Sequences

* Definition: A sequence $\{a_n\}$ converges to a real number $L$ (its limit) if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $n \ge N$, $|a_n - L| < \epsilon$. This is denoted as $\lim_{n \to \infty} a_n = L$.
* Uniqueness of Limit: If a sequence converges, its limit is unique.
* Boundedness of Convergent Sequences: Every convergent sequence is bounded. (The converse is false; e.g., $\{(-1)^n\}$ is bounded but not convergent).

Divergent Sequences

* Diverging to $\infty$: A sequence $\{a_n\}$ diverges to $\infty$ if for every $M > 0$, there exists $N \in \mathbb{N}$ such that for all $n \ge N$, $a_n > M$. Denoted $\lim_{n \to \infty} a_n = \infty$.
* Diverging to $-\infty$: A sequence $\{a_n\}$ diverges to $-\infty$ if for every $M < 0$, there exists $N \in \mathbb{N}$ such that for all $n \ge N$, $a_n < M$. Denoted $\lim_{n \to \infty} a_n = -\infty$.
* Oscillating Sequences: A sequence that is neither convergent nor divergent to $\pm \infty$ is called an oscillating sequence (e.g., $\{(-1)^n\}$).

Monotonic Sequences

* Monotonically Increasing: $a_n \le a_{n+1}$ for all $n \in \mathbb{N}$.
* Strictly Increasing: $a_n < a_{n+1}$ for all $n \in \mathbb{N}$.
* Monotonically Decreasing: $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
* Strictly Decreasing: $a_n > a_{n+1}$ for all $n \in \mathbb{N}$.
* Monotonic Sequence: A sequence that is either monotonically increasing or monotonically decreasing.
* Monotone Convergence Theorem (MCT):
* Every monotonically increasing sequence that is bounded above converges.
* Every monotonically decreasing sequence that is bounded below converges.
* (Equivalently, a monotonic sequence converges if and only if it is bounded.)

Subsequences

* Definition: Given a sequence $\{a_n\}$, if $\{n_k\}$ is a strictly increasing sequence of natural numbers ($n_1 < n_2 < n_3 < \dots$), then $\{a_{n_k}\}$ is called a subsequence of $\{a_n\}$.
* Properties:
* If a sequence $\{a_n\}$ converges to $L$, then every subsequence $\{a_{n_k}\}$ also converges to $L$.
* If a sequence has two subsequences that converge to different limits, then the sequence itself diverges.
* Limit Point (Cluster Point): A real number $L$ is a limit point of a sequence $\{a_n\}$ if there exists a subsequence of $\{a_n\}$ that converges to $L$.
* Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence. (This implies every bounded sequence has at least one limit point).

Cauchy Sequences

* Definition: A sequence $\{a_n\}$ is a Cauchy sequence if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $m, n \ge N$, $|a_m - a_n| < \epsilon$.
* Cauchy's General Principle of Convergence: A sequence of real numbers converges if and only if it is a Cauchy sequence. (This theorem establishes the completeness of the real number system).
* Properties: Every Cauchy sequence is bounded.

Algebra of Limits

If $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = M$, then:
* $\lim_{n \to \infty} (a_n \pm b_n) = L \pm M$
* $\lim_{n \to \infty} (c \cdot a_n) = c \cdot L$ for any constant $c \in \mathbb{R}$
* $\lim_{n \to \infty} (a_n \cdot b_n) = L \cdot M$
* $\lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = \frac{L}{M}$, provided $M \ne 0$ and $b_n \ne 0$ for all $n$.
* If $a_n \le b_n$ for all $n$, then $L \le M$.

Standard Limits

These are frequently used limits:
* $\lim_{n \to \infty} \frac{1}{n^p} = 0$ for $p > 0$.
* $\lim_{n \to \infty} x^{1/n} = 1$ for $x > 0$.
* $\lim_{n \to \infty} n^{1/n} = 1$.
* $\lim_{n \to \infty} x^n = 0$ if $|x| < 1$.
* $\lim_{n \to \infty} x^n = \infty$ if $x > 1$.
* $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$.
* $\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x$.
* $\lim_{n \to \infty} \frac{x^n}{n!} = 0$ for any $x \in \mathbb{R}$.
* $\lim_{n \to \infty} \frac{\log n}{n} = 0$.

Squeeze (Sandwich) Theorem

If $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ are sequences such that $a_n \le b_n \le c_n$ for all $n \ge N_0$ (for some $N_0 \in \mathbb{N}$), and if $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$.

Cauchy's First Theorem on Limits

If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$.

Cauchy's Second Theorem on Limits

If $\{a_n\}$ is a sequence of positive real numbers such that $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then $\lim_{n \to \infty} (a_n)^{1/n} = L$.

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Important Formulas

Definition of Convergence:

Algebra of Limits:

If $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = M$:

Cauchy Sequence Definition:

Standard Limits:

Cauchy's Second Theorem on Limits:

If $a_n > 0$ and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then:

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Examples

Example: Limit of a sequence using algebraic manipulation

Find the value of $\lim_{n \to \infty} (\sqrt{4n^2 + n} - 2n)$.

Example: Limit using Squeeze Theorem

Find the value of $\lim_{n \to \infty} \left[ \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right]$.
Let $S_n = \sum_{k=1}^{n} \frac{1}{\sqrt{n^2+k}}$.
For each term $\frac{1}{\sqrt{n^2+k}}$, we have $n^2+1 \le n^2+k \le n^2+n$.
Taking reciprocals and square roots:

Summing $n$ terms:



By the Squeeze Theorem, $\lim_{n \to \infty} S_n = 1$.

Example: Limit using Cauchy's Second Theorem on Limits

The value of $\lim_{n \to \infty} \left[\frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n\right]^{1/n}$.
Let $a_n = \prod_{k=1}^{n} \left(\frac{k+1}{k}\right)^k$. We need to find $\lim_{n \to \infty} (a_n)^{1/n}$.
Using Cauchy's Second Theorem on Limits, we evaluate $\lim_{n \to \infty} \frac{a_{n+1}}{a_n}$.


By Cauchy's Second Theorem on Limits, $\lim_{n \to \infty} (a_n)^{1/n} = e$.

Example: Non-convergent sequence

The sequence $\left\{(-1)^n \cdot \frac{n+1}{n}\right\}$.
Let $a_n = (-1)^n \cdot \frac{n+1}{n}$.
Consider subsequences:
* For even $n=2k$: $a_{2k} = (-1)^{2k} \cdot \frac{2k+1}{2k} = 1 + \frac{1}{2k}$. $\lim_{k \to \infty} a_{2k} = 1$.
* For odd $n=2k-1$: $a_{2k-1} = (-1)^{2k-1} \cdot \frac{2k-1+1}{2k-1} = -1 \cdot \frac{2k}{2k-1} = -1 \left(1 + \frac{1}{2k-1}\right)$. $\lim_{k \to \infty} a_{2k-1} = -1$.
Since the sequence has two subsequences converging to different limits ($1$ and $-1$), the sequence is not convergent. It is an oscillating sequence.

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Important Points/Tips for Exam Preparation

Master Definitions: Understand the $\epsilon-N$ definition of convergence and Cauchy sequences. While direct proofs might be rare, conceptual understanding is vital for MCQs.

Standard Limits: Memorize and be able to quickly apply standard limits (e.g., $\lim_{n \to \infty} n^{1/n}=1$, $\lim_{n \to \infty} (1+x/n)^n=e^x$). These are frequent in exams.

Limit Theorems: Be proficient in applying the algebra of limits, Squeeze Theorem, Cauchy's First and Second Theorems on Limits. These are crucial for evaluating complex limits.

Monotone Convergence Theorem (MCT): Understand that a monotonic sequence converges if and only if it is bounded. This is a powerful tool for determining convergence without finding the exact limit.

Bolzano-Weierstrass Theorem: Remember that every bounded sequence has a convergent subsequence. This helps in understanding limit points and non-convergence.

Cauchy's General Principle of Convergence: A sequence converges if and only if it is a Cauchy sequence. This provides an intrinsic test for convergence.

Divergence Criteria: A sequence diverges if it is unbounded, or if it has two subsequences converging to different limits (oscillating).

Indeterminate Forms: Practice handling indeterminate forms ($\infty - \infty$, $0 \cdot \infty$, $1^\infty$, etc.) using algebraic manipulation (rationalization, factoring) or logarithms.

PYQ Analysis: The provided PYQs emphasize calculating limits of various sequences, identifying convergent/divergent sequences, and applying theorems like the Squeeze Theorem or Cauchy's Second Theorem. Focus your practice accordingly.

Practice: Solve a wide variety of problems, especially those involving square roots, fractions, powers, and sums, to build speed and accuracy.

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Sequences of Real Numbers: Part 3 - Advanced Topics

Chapter Overview

This section delves into advanced concepts and techniques for analyzing sequences of real numbers, crucial for tackling complex problems in competitive exams like CUET PG. We will cover sophisticated methods for evaluating limits, understanding the deeper properties of convergent and divergent sequences, and exploring special types of sequences and theorems. Mastery of these topics requires a strong foundation in basic sequence definitions, boundedness, and monotonicity.

Key Concepts

Subsequences

* A subsequence of a sequence $\{a_n\}$ is a sequence $\{a_{n_k}\}$ where $n_1 < n_2 < n_3 < \dots$ are strictly increasing positive integers.
* Theorem: If a sequence $\{a_n\}$ converges to $L$, then every subsequence $\{a_{n_k}\}$ also converges to $L$.
* Theorem: If a sequence $\{a_n\}$ has two subsequences that converge to different limits, or if one subsequence diverges, then the original sequence $\{a_n\}$ diverges. This is particularly useful for showing divergence of oscillating sequences.
* Example: The sequence $\{(-1)^n\}$ has subsequences $\{a_{2k}\} = \{1\}$ (converges to 1) and $\{a_{2k-1}\} = \{-1\}$ (converges to -1). Since they converge to different limits, $\{(-1)^n\}$ diverges.

Cauchy Sequences

* A sequence $\{a_n\}$ is called a Cauchy sequence if for every $\epsilon > 0$, there exists a positive integer $N$ such that for all $m, n > N$, we have $|a_m - a_n| < \epsilon$.
* Cauchy's Convergence Criterion: A sequence of real numbers converges if and only if it is a Cauchy sequence. This theorem establishes the completeness of the real number system.
* Properties:
* Every convergent sequence is a Cauchy sequence.
* Every Cauchy sequence is bounded.
* Every Cauchy sequence of real numbers converges.

Advanced Limit Evaluation Techniques

* Squeeze Theorem (Sandwich Theorem)
* If $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ are sequences such that $a_n \le b_n \le c_n$ for all $n > N_0$ (for some $N_0$), and $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$.
* Application: Useful for limits involving sums or terms that are difficult to simplify directly.
* Example (PYQ type): To evaluate $\lim_{n \to \infty} \left[ \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right]$.
* Each term $\frac{1}{\sqrt{n^2+k}}$ satisfies $\frac{1}{\sqrt{n^2+n}} \le \frac{1}{\sqrt{n^2+k}} \le \frac{1}{\sqrt{n^2+1}}$ for $k \in \{1, \dots, n\}$.
* Summing $n$ terms: $n \cdot \frac{1}{\sqrt{n^2+n}} \le \sum_{k=1}^n \frac{1}{\sqrt{n^2+k}} \le n \cdot \frac{1}{\sqrt{n^2+1}}$.
* $\lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+1/n}} = 1$.
* $\lim_{n \to \infty} \frac{n}{\sqrt{n^2+1}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+1/n^2}} = 1$.
* By Squeeze Theorem, the limit is $1$.

* Cauchy's First Theorem on Limits
* If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$.
* Note: The converse is not necessarily true.

* Cauchy's Second Theorem on Limits
* If $\{a_n\}$ is a sequence of positive real numbers and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then $\lim_{n \to \infty} (a_n)^{1/n} = L$.
* Application: Very useful for limits of $n$-th roots, especially when direct evaluation is difficult.
* Example (PYQ type): To evaluate $\lim_{n \to \infty} \sqrt[n]{n}$.
* Let $a_n = n$. Then $\frac{a_{n+1}}{a_n} = \frac{n+1}{n} = 1 + \frac{1}{n}$.
* $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} (1 + \frac{1}{n}) = 1$.
* By Cauchy's Second Theorem, $\lim_{n \to \infty} \sqrt[n]{n} = 1$.
* Example (PYQ type): To evaluate $\lim_{n \to \infty} \left[\frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n\right]^{1/n}$.
* Let $x_n = \frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n$. We need $\lim_{n \to \infty} (x_n)^{1/n}$.
* Let $a_n = \left(\frac{n+1}{n}\right)^n$. Then $x_n = \prod_{k=1}^n a_k$. This is not directly $a_n^{1/n}$.
* This is a product of terms, $P_n = \prod_{k=1}^n b_k$. We need $\lim_{n \to \infty} (P_n)^{1/n}$.
* If $\lim_{n \to \infty} b_n = L$, then $\lim_{n \to \infty} (b_1 b_2 \dots b_n)^{1/n} = L$.
* Here $b_k = \left(\frac{k+1}{k}\right)^k$.
* $\lim_{k \to \infty} b_k = \lim_{k \to \infty} \left(1 + \frac{1}{k}\right)^k = e$.
* Therefore, the limit is $e$.

* L'Hopital's Rule (for sequences)
* While L'Hopital's Rule is for functions, it can be applied to sequences by considering $f(x)$ such that $f(n) = a_n$. If $\lim_{x \to \infty} f(x)$ is of indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}$. If this limit exists, then $\lim_{n \to \infty} a_n$ also exists and is equal to it.
* Caution: Ensure the function $f(x)$ is differentiable and the conditions for L'Hopital's Rule are met.

* Using Logarithms for Indeterminate Forms ($1^\infty, 0^0, \infty^0$)
* If $\lim_{n \to \infty} a_n^{b_n}$ is an indeterminate form, let $L = \lim_{n \to \infty} a_n^{b_n}$.
* Then $\ln L = \lim_{n \to \infty} \ln(a_n^{b_n}) = \lim_{n \to \infty} b_n \ln a_n$.
* This often converts the indeterminate form into $0 \cdot \infty$ or $\infty \cdot 0$, which can then be rewritten as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ for L'Hopital's Rule.
* Finally, $L = e^{\ln L}$.

* Algebraic Manipulation (Conjugates)
* For indeterminate forms like $\infty - \infty$, multiplying by the conjugate is a common technique.
* Example (PYQ type): To evaluate $\lim_{n \to \infty} (\sqrt{4n^2 + n} - 2n)$.
* Multiply by conjugate:

* Telescoping Products/Sums
* A product (or sum) is telescoping if intermediate terms cancel out.
* Example (PYQ type): To evaluate $S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right)$.
* Each term can be factored: $1 - \frac{1}{k^2} = \frac{k^2-1}{k^2} = \frac{(k-1)(k+1)}{k \cdot k}$.
* The product becomes:

* Notice the cancellations:

* So, $S = \lim_{n \to \infty} \left(\frac{1}{2} \cdot \frac{n+1}{n}\right) = \frac{1}{2} \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = \frac{1}{2} \cdot 1 = \frac{1}{2}$.

Important Formulas

Cauchy's Convergence Criterion:

A sequence $\{a_n\}$ converges if and only if for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $m, n > N$, $|a_m - a_n| < \epsilon$.

Squeeze Theorem:

If $a_n \le b_n \le c_n$ for $n > N_0$ and $\lim_{n \to \infty} a_n = L$, $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$.

Cauchy's First Theorem on Limits:

If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$.

Cauchy's Second Theorem on Limits:

If $\{a_n\}$ is a sequence of positive real numbers and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then $\lim_{n \to \infty} (a_n)^{1/n} = L$.
Also, if $\lim_{n \to \infty} b_n = L$, then $\lim_{n \to \infty} (b_1 b_2 \dots b_n)^{1/n} = L$.

Special Limits:

*
*
*
*
*
*
*

Important Points/Tips for Exam Preparation

Master Indeterminate Forms: Be proficient in recognizing and resolving indeterminate forms like $\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 1^\infty, 0^0, \infty^0$. Each requires specific techniques (L'Hopital's, conjugates, logarithms).

Understand Theoretical Concepts: Questions on Assertion-Reason type often test the fundamental definitions and theorems. For example, knowing that "Every monotonic sequence is convergent" is FALSE (it must also be bounded) is crucial. A sequence like $\{n\}$ is monotonic but diverges.

Divergence Criteria: Remember that if a sequence has two subsequences converging to different limits, or if it is unbounded, it diverges. This is key for sequences like $\{(-1)^n \frac{n+1}{n}\}$.

Practice Special Limits: Memorize and understand the derivation of common limits like $\lim_{n \to \infty} n^{1/n}$ and $\lim_{n \to \infty} (1 + 1/n)^n$.

Identify the Right Tool: For limits of sums, consider the Squeeze Theorem. For limits of $n$-th roots or products, Cauchy's Second Theorem on Limits is often the most efficient method.

Pattern Recognition Questions: While not strictly "sequences of real numbers" in a mathematical analysis sense, CUET PG often includes numerical sequence pattern questions (e.g., $11, 20, 38, 74, \_\_\_\_\_\_\_\_$). These test logical reasoning. Practice identifying arithmetic progressions, geometric progressions, differences of differences, or combinations of operations.

Review Basic Properties: Ensure a solid understanding of boundedness, monotonicity, and the algebra of limits, as these form the basis for advanced topics.

Work Through PYQs: The provided PYQs are excellent examples of the types of problems you can expect. Practice similar problems to build confidence and speed.

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Sequences of Real Numbers: Part 4 - Applications

Chapter Overview

This section focuses on applying the theoretical concepts of sequences of real numbers to solve various problems. It covers techniques for evaluating limits of complex sequences, determining convergence or divergence, and recognizing patterns within sequences. Mastery of these applications is crucial for the CUET PG examination.

Key Concepts

* Indeterminate Forms: Expressions such as $\frac{0}{0}$, $\frac{\infty}{\infty}$, $\infty - \infty$, $0 \cdot \infty$, $1^\infty$, $0^0$, and $\infty^0$ that require specific algebraic or calculus-based techniques for evaluation. * Algebraic Manipulation: Techniques like multiplying by the conjugate, dividing by the highest power of $n$, or factoring to simplify expressions and resolve indeterminate forms. * Squeeze Theorem (Sandwich Theorem): If $a_n \le b_n \le c_n$ for all $n > N$ (for some integer $N$), and $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$. * Cauchy's First Theorem on Limits: If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$. * Cauchy's Second Theorem on Limits (for $n$-th root of a product): If $a_n > 0$ for all $n$ and $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} (a_1 a_2 \dots a_n)^{1/n} = L$. (A common variant states: If $a_n > 0$ and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then $\lim_{n \to \infty} (a_n)^{1/n} = L$.) * Standard Limits: Fundamental limits that are frequently used in evaluations. * Monotonicity and Boundedness: A sequence that is both monotonic (either non-decreasing or non-increasing) and bounded (both above and below) is guaranteed to be convergent. * Oscillating Sequences: Sequences that do not converge to a single limit and do not diverge to $\pm \infty$. They typically oscillate between multiple values or within a finite range.

Important Formulas

* * * * * * *

Examples

Example 1: Limit of $\infty - \infty$ form (using conjugate)
Evaluate $\lim_{n \to \infty} (\sqrt{4n^2 + n} - 2n)$.
Solution:

Example 2: Limit using Squeeze Theorem
Evaluate $\lim_{n \to \infty} \left[ \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right]$.
Solution:
Let $S_n = \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}}$.
There are $n$ terms in the sum.
The smallest term in the sum is $\frac{1}{\sqrt{n^2+n}}$ and the largest term is $\frac{1}{\sqrt{n^2+1}}$.
Thus, we can bound $S_n$:

Now, evaluate the limits of the lower and upper bounds:


By the Squeeze Theorem, since both bounds converge to $1$, $\lim_{n \to \infty} S_n = 1$.

Example 3: Limit of $n$-th root
Evaluate $\lim_{n \to \infty} \sqrt[n]{n}$.
Solution:
Let $L = \lim_{n \to \infty} n^{1/n}$.
Take the natural logarithm of both sides:

This is an $\frac{\infty}{\infty}$ indeterminate form. Applying L'Hopital's Rule (treating $n$ as a continuous variable $x$):

So, $\ln L = 0$, which implies $L = e^0 = 1$.
Thus, $\lim_{n \to \infty} \sqrt[n]{n} = 1$.

Example 4: Limit of a product using Cauchy's Second Theorem
Evaluate $\lim_{n \to \infty} \left[\frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n\right]^{1/n}$.
Solution:
Let the sequence be $P_n = \frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n$. We need to find $\lim_{n \to \infty} (P_n)^{1/n}$.
This expression is of the form $\lim_{n \to \infty} (x_1 x_2 \dots x_n)^{1/n}$, where $x_k = \left(\frac{k+1}{k}\right)^k$.
By Cauchy's Second Theorem on Limits, if $\lim_{n \to \infty} x_n = L$, then $\lim_{n \to \infty} (x_1 x_2 \dots x_n)^{1/n} = L$.
Let's evaluate $\lim_{n \to \infty} x_n$:

Therefore, by Cauchy's Second Theorem on Limits, the given limit is $e$.

Example 5: Limit of a telescoping product
Evaluate $S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right)$.
Solution:
Consider the general term $1 - \frac{1}{k^2}$:

Let $P_n = \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right)$.


Rearranging the terms to observe the cancellation:


Now, take the limit:

Example 6: Convergence of an oscillating sequence
Determine if the sequence $\left\{(-1)^n \cdot \frac{n+1}{n}\right\}$ is convergent.
Solution:
Let $a_n = (-1)^n \cdot \frac{n+1}{n}$.
We can rewrite $\frac{n+1}{n} = 1 + \frac{1}{n}$.
As $n \to \infty$, $\frac{1}{n} \to 0$, so $\frac{n+1}{n} \to 1$.
The term $(-1)^n$ alternates between $-1$ and $1$.
Consider the subsequences for even and odd $n$:
For even $n = 2k$:

For odd $n = 2k-1$:


Since the sequence has two different limit points ($1$ and $-1$), it is not convergent. It is an oscillating sequence.

Example 7: Pattern Recognition (Aptitude type)
What should come in place of the question mark? $11, 20, 38, 74, \_\_\_\_\_\_\_\_$?
Solution:
Let's find the differences between consecutive terms:
$20 - 11 = 9$
$38 - 20 = 18$
$74 - 38 = 36$
The differences are $9, 18, 36$. This is a geometric progression where each term is doubled ($9 \times 2 = 18$, $18 \times 2 = 36$).
The next difference should be $36 \times 2 = 72$.
So, the next term in the sequence is $74 + 72 = 146$.

Important Points/Tips for Exam Preparation

* Master Indeterminate Forms: Be proficient in identifying and resolving all types of indeterminate forms using algebraic manipulation, L'Hopital's Rule (when applicable for functions, then extended to sequences), or by recognizing standard limits. * Know Standard Limits: Memorize or be able to quickly derive common limits like $\lim_{n \to \infty} n^{1/n}$, $\lim_{n \to \infty} (1+1/n)^n$, etc. These are frequently tested. * Practice Squeeze Theorem: This theorem is a powerful tool for evaluating limits of sums or terms that can be bounded by other sequences. The key is to construct appropriate bounding sequences. * Understand Cauchy's Theorems: These theorems provide elegant shortcuts for evaluating limits of arithmetic means and $n$-th roots of products. * Convergence Criteria: Always check for monotonicity and boundedness when determining the convergence of a sequence. Remember that oscillating sequences do not converge. * Subsequences: Utilize subsequences (e.g., considering even and odd terms separately) to prove divergence or to find limit points of a sequence. * Pattern Recognition: For aptitude-style questions, look for arithmetic, geometric, or mixed patterns in the terms themselves or in the differences/ratios between consecutive terms. Practice various types of sequence puzzles. * Careful with Algebra: Many errors in limit evaluation arise from algebraic mistakes, especially when dealing with square roots, fractions, and exponents. Double-check each step of your calculations.

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Chapter: Sequences of Real Numbers - Part 5: Summary

Key Points

Definition of a Sequence: A sequence of real numbers is a function $f: \mathbb{N} \to \mathbb{R}$, denoted by $\{a_n\}_{n=1}^\infty$ or $(a_n)$.

Convergence of a Sequence:

* A sequence $\{a_n\}$ converges to a limit $L \in \mathbb{R}$ if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $n \ge N$, $|a_n - L| < \epsilon$. We write $\lim_{n \to \infty} a_n = L$.
* A sequence that does not converge is said to diverge.
* Uniqueness of Limit: If a sequence converges, its limit is unique.

Boundedness:

* A sequence $\{a_n\}$ is bounded above if there exists $M \in \mathbb{R}$ such that $a_n \le M$ for all $n \in \mathbb{N}$.
* A sequence $\{a_n\}$ is bounded below if there exists $m \in \mathbb{R}$ such that $a_n \ge m$ for all $n \in \mathbb{N}$.
* A sequence is bounded if it is both bounded above and bounded below (i.e., there exists $K > 0$ such that $|a_n| \le K$ for all $n \in \mathbb{N}$).
* Theorem: Every convergent sequence is bounded. (The converse is not true, e.g., $\{(-1)^n\}$).

Algebra of Limits: If $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = M$, then:

*
*
*
*

Monotonic Sequences:

* A sequence $\{a_n\}$ is monotonically increasing if $a_{n+1} \ge a_n$ for all $n$.
* A sequence $\{a_n\}$ is monotonically decreasing if $a_{n+1} \le a_n$ for all $n$.
* A sequence is monotonic if it is either monotonically increasing or monotonically decreasing.
* Monotone Convergence Theorem (MCT): A monotonic sequence converges if and only if it is bounded.
* A monotonically increasing sequence converges if it is bounded above.
* A monotonically decreasing sequence converges if it is bounded below.

Squeeze Theorem (Sandwich Theorem): If $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ are sequences such that $a_n \le b_n \le c_n$ for all $n \ge N_0$ (for some $N_0 \in \mathbb{N}$), and $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$.

* Example: To evaluate $\lim_{n \to \infty} \left[ \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} \right]$:
For each term $\frac{1}{\sqrt{n^2+k}}$, we have $\frac{1}{\sqrt{n^2+n}} \le \frac{1}{\sqrt{n^2+k}} \le \frac{1}{\sqrt{n^2+1}}$ for $k=1, \dots, n$.
Summing $n$ terms:

As $n \to \infty$, $\lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+1/n}} = 1$ and $\lim_{n \to \infty} \frac{n}{\sqrt{n^2+1}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+1/n^2}} = 1$.
By Squeeze Theorem, the limit is $1$.

Cauchy's First Theorem on Limits: If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$.

Cauchy's Second Theorem on Limits (and its variants):

* If $\{a_n\}$ is a sequence of positive real numbers and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, then $\lim_{n \to \infty} (a_n)^{1/n} = L$.
* Variant for Products: If $\{a_n\}$ is a sequence of positive real numbers and $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} (a_1 a_2 \dots a_n)^{1/n} = L$.
* Example: To evaluate $\lim_{n \to \infty} \left[\frac{2}{1} \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \dots \left(\frac{n+1}{n}\right)^n\right]^{1/n}$:
Let $b_k = \left(\frac{k+1}{k}\right)^k$. Then $\lim_{k \to \infty} b_k = \lim_{k \to \infty} \left(1+\frac{1}{k}\right)^k = e$.
The expression is $\lim_{n \to \infty} (b_1 b_2 \dots b_n)^{1/n}$. By the variant theorem, this limit is $e$.

Stolz-Cesaro Theorem: If $\{a_n\}$ and $\{b_n\}$ are sequences such that $\{b_n\}$ is strictly monotonic and $\lim_{n \to \infty} b_n = \infty$ (or $-\infty$), and if $\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L$, then $\lim_{n \to \infty} \frac{a_n}{b_n} = L$.

Standard Limits to Remember:

*
*
*
*
*
*
*
*

Cauchy Sequences:

* A sequence $\{a_n\}$ is a Cauchy sequence if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $m, n \ge N$, $|a_m - a_n| < \epsilon$.
* Cauchy's General Principle of Convergence: A sequence of real numbers converges if and only if it is a Cauchy sequence.

Divergence Criteria:

* A sequence diverges if it is unbounded.
* A sequence diverges if it has two subsequences that converge to different limits (e.g., $\{(-1)^n\}$ has subsequences converging to $1$ and $-1$).
* A sequence diverges if it oscillates (e.g., $\{(-1)^n n\}$).

Techniques for Evaluating Limits:

* Rationalization: Useful for expressions involving square roots.
* Example: $\lim_{n \to \infty} (\sqrt{4n^2 + n} - 2n)$



* Telescoping Products: For products that simplify by cancellation.
* Example: $\lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right)$
Each term $1 - \frac{1}{k^2} = \frac{k^2-1}{k^2} = \frac{(k-1)(k+1)}{k \cdot k}$.
The product $P_n$ is: