Sequences and Convergence

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# Sequences and Convergence ## 🎯 Learning Objectives By the end of this topic, you will be able to: 1. **Define** a sequence and distinguish it from a function 2.

Sequences and Convergence

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define a sequence and distinguish it from a function
Compute limits of sequences and determine convergence/divergence
Identify monotonic and bounded sequences
Recognize standard limits and their applications

📋 Prerequisites

Functions — function notation, domain, range
Exponent rules — handling powers and fractions

📖 Core Content

10.1 What is a Sequence?

10.1.1 Intuition

A sequence is a list of numbers in a specific order. Like a playlist: track 1, track 2, track 3, ... Each position has a value. Unlike a set, order matters and repetition is allowed:

(1, 2, 2, 3)

is a valid sequence.

💡 Why this matters: Sequences model discrete processes: daily stock prices, CPU temperature readings, iterations of an algorithm. In ML, training loss over epochs is a sequence.

10.1.2 Definition

A sequence is a function from

\mathbb{N}

\mathbb{R}

a: \mathbb{N} \to \mathbb{R}

. We write

a_n

for the

n

-th term, and

\{a_n\}_{n=1}^{\infty}

(a_n)

for the whole sequence. Examples:

$a_n = \frac{1}{n}$ : $(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots)$
$a_n = n^2$ : $(1, 4, 9, 16, \ldots)$
$a_n = (-1)^n$ : $(-1, 1, -1, 1, \ldots)$
$a_n = \frac{n}{n+1}$ : $(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots)$

10.2 Limit of a Sequence

10.2.1 Intuition

Does the sequence get arbitrarily close to some number

L

n

gets large? If so,

L

is the limit.

$\frac{1}{n}$ gets close to $0$ : limit $= 0$
$n^2$ grows without bound: no finite limit (diverges to $\infty$ )
$(-1)^n$ oscillates: no limit

🔍 Formal definition (epsilon-N): $\lim_{n\to\infty} a_n = L$ if for every $\epsilon > 0$ , there exists $N \in \mathbb{N}$ such that for all $n > N$ , $|a_n - L| < \epsilon$ .

10.2.2 Convergence vs. Divergence

Behavior	Term	Example
Approaches a finite $L$	Converges to $L$	$a_n = \frac{1}{n} \to 0$
Grows without bound	Diverges to $\infty$	$a_n = n \to \infty$
Oscillates without settling	Diverges (no limit)	$a_n = (-1)^n$
Settles to a pattern	May converge or diverge	$a_n = \sin n$ diverges

10.2.3 Standard Limits

\lim_{n\to\infty} \frac{1}{n^p} &= 0 && (p > 0) \\ \lim_{n\to\infty} r^n &= 0 && (|r| < 1) \\ \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n &= e \\ \lim_{n\to\infty} \sqrt[n]{n} &= 1 \\ \lim_{n\to\infty} \frac{n^k}{b^n} &= 0 && (b > 1, \text{ exponentials beat polynomials}) \end{aligned}

#### 10.2.4 Limit Laws If $\\lim a_n = L$ and $\\lim b_n = M$: - $\\lim(a_n + b_n) = L + M$ - $\\lim(a_n - b_n) = L - M$ - $\\lim(a_n \\cdot b_n) = L \\cdot M$ - $\\lim(a_n / b_n) = L / M$ (if $M \\neq 0$) - $\\lim(c \\cdot a_n) = c \\cdot L$ #### 10.2.5 Worked Examples **Example 1.1 (Easy):** Find $\\lim_{n\\to\\infty} \\frac{2n + 3}{n + 1}$. \\lim_{n\\to\\infty} \\frac{2n + 3}{n + 1} = \\lim_{n\\to\\infty} \\frac{2 + \\frac{3}{n}}{1 + \\frac{1}{n}} = \\frac{2 + 0}{1 + 0} = 2 \\boxed{2} **Example 1.2 (Medium):** Find $\\lim_{n\\to\\infty} \\frac{n^2 + 2n}{3n^2 - n + 1}$. \\lim_{n\\to\\infty} \\frac{n^2 + 2n}{3n^2 - n + 1} = \\lim_{n\\to\\infty} \\frac{1 + \\frac{2}{n}}{3 - \\frac{1}{n} + \\frac{1}{n^2}} = \\frac{1}{3} \\boxed{\\frac{1}{3}} **Example 1.3 (Hard):** Find $\\lim_{n\\to\\infty} \\left(\\sqrt{n+1} - \\sqrt{n}\\right)$.

\begin{aligned} \sqrt{n+1} - \sqrt{n} &= \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} \\ &= \frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}} \to 0 \end{aligned}

\\boxed{0} \### 10.3 Types of Sequences | Type | Definition | Example | |:---|:---|:---| | **Increasing** | $a_{n+1} \\geq a_n$ for all $n$ | $a_n = n$ | | **Strictly increasing** | $a_{n+1} > a_n$ | $a_n = n^2$ | | **Decreasing** | $a_{n+1} \\leq a_n$ | $a_n = 1/n$ | | **Monotonic** | Either increasing or decreasing | $a_n = \\frac{n}{n+1}$ | | **Bounded above** | $a_n \\leq M$ for some $M$ | $a_n = \\frac{n}{n+1} \\leq 1$ | | **Bounded below** | $a_n \\geq m$ for some $m$ | $a_n = n \\geq 1$ | | **Bounded** | Both above and below | $a_n = (-1)^n$ is bounded by $\[-1,1\]$ | **Monotone Convergence Theorem:** Every bounded monotonic sequence converges. --- ## 📐 Key Formulas — Summary Table | Concept | Formula/Result | Notes | |:---|:---|:---| | Sequence | $a_n = f(n)$, $n \\in \\mathbb{N}$ | Ordered list | | Limit | $\\lim_{n\\to\\infty} a_n = L$ | Gets arbitrarily close to $L$ | | $1/n^p$ | $\\to 0$ if $p > 0$ | Standard limit | | $r^n$ | $\\to 0$ if $|r| < 1$ | Geometric sequence | | $(1+1/n)^n$ | $\\to e$ | Definition of $e$ | | Ratio test | Poly/exp $\\to 0$ | Exponential dominates | | Squeeze theorem | If $b_n \\leq a_n \\leq c_n$ and $b_n,c_n \\to L$, then $a_n \\to L$ | | | MCT | Bounded + monotonic → convergent | | --- ## ⚠️ Common Pitfalls ### Pitfall 1: Confusing Sequence with Series A sequence is a list; a series is a sum of terms. Sequences converge if terms approach a limit. Series converge if the sum of infinitely many terms is finite. ### Pitfall 2: Thinking Bounded Means Convergent $(-1)^n$ is bounded between $-1$ and $1$ but does NOT converge (it oscillates). Bounded + monotonic → convergent. ### Pitfall 3: Incorrectly Applying Limit Laws to $0/0$ or $\\infty/\\infty$ Forms Must use algebraic manipulation (factor, rationalize, divide by highest power) before applying laws. --- ## 📝 Practice Questions > **Q1: Write the first 4 terms of $a_n = \\frac{n}{n+1}$.** > > $n=1: 1/2$, $n=2: 2/3$, $n=3: 3/4$, $n=4: 4/5$ > >

\boxed{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}}

> **Q2: Find $\\lim_{n\\to\\infty} \\frac{3n+2}{2n-1}$.** > > Divide by $n$: $\\frac{3 + 2/n}{2 - 1/n} \\to \\frac{3}{2}$ > >

\boxed{\frac{3}{2}}

> **Q3: Does $a_n = \\frac{n^2 + 1}{n}$ converge?** > > $a_n = n + \\frac{1}{n} \\to \\infty$, so diverges. > >

\boxed{\text{Diverges to } \infty}

> **Q4: Determine if $a_n = \\frac{(-1)^n}{n}$ converges.** > > $-\\frac{1}{n} \\leq a_n \\leq \\frac{1}{n}$, both bounds $\\to 0$, so by squeeze theorem $a_n \\to 0$. > >

\boxed{\text{Converges to } 0}

> **Q5: Find $\\lim_{n\\to\\infty} \\frac{n}{\\sqrt{n^2 + 1}}$.** > > Divide numerator and denominator by $n$: $\\frac{1}{\\sqrt{1 + 1/n^2}} \\to \\frac{1}{\\sqrt{1}} = 1$ > >

\boxed{1}

> **Q6: Find $\\lim_{n\\to\\infty} \\frac{2^n}{3^n}$.** > > $(2/3)^n$, since $|2/3| < 1$, limit $= 0$. > >

\boxed{0}

> **Q7: Is $a_n = \\frac{1}{2^n}$ increasing or decreasing?** > > $a_1 = 1/2$, $a_2 = 1/4$, $a_3 = 1/8$ — decreasing. > >

\boxed{\text{Decreasing}}

> **Q8: Find $\\lim_{n\\to\\infty} \\frac{\\ln n}{n}$.** > > As $n \\to \\infty$, $\\ln n$ grows slower than $n$, so limit $= 0$. > >

\boxed{0}

> **Q9: Find $\\lim_{n\\to\\infty} \\frac{n^3}{2^n}$.** > > Exponential dominates polynomial → limit $= 0$. > >

\boxed{0}

> **Q10: Evaluate $\\lim_{n\\to\\infty} \\left(1 + \\frac{2}{n}\\right)^n$.** > > $\\left(1 + \\frac{2}{n}\\right)^n = \\left\[\\left(1 + \\frac{2}{n}\\right)^{n/2}\\right\]^2 \\to e^2$ as $n \\to \\infty$. > >

\boxed{e^2}$$ --- ## 🔗 Cross-References - Next topic: [Limits of Functions](/courses/bsma1001/notes/limits-functions) - Related: [Continuity](/courses/bsma1001/notes/limits-continuity) - Across courses: BSMA1003 Maths 2 (series, convergence tests) Join Discord Previous6.2 Exponential & Log EquationsNext7.2 Limits of Functions