Section 11.1: Sequences

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MATH163

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What is This Section About?

Welcome to the fascinating world of sequences! In this section, we explore one of the foundational concepts in calculus and mathematical analysis. A sequence is simply an ordered list of numbers, generated by some rule or formula. You’ve likely encountered sequences before—perhaps in the form of the counting numbers (1, 2, 3, 4, …), or in patterns like the Fibonacci sequence (1, 1, 2, 3, 5, 8, …).

But why do sequences matter? Sequences appear throughout mathematics and real-world applications:

• Newton’s Method uses sequences to approximate solutions to equations
• Zeno’s Paradox explores the philosophical implications of infinite sequences
• Series and Convergence Tests (coming in later sections) build directly on sequence foundations
• Iterative Algorithms in computer science rely on sequence behavior

Understanding sequences is your gateway to deeper topics in analysis, numerical methods, and mathematical modeling.

The Big Picture: What You’ll Learn

By the end of this section, you’ll be able to:

1. Define and work with sequences using both explicit formulas and recursive definitions
2. Determine convergence or divergence of sequences using rigorous definitions and techniques
3. Apply limit laws and the Squeeze Theorem to evaluate sequence limits
4. Analyze recursive sequences and find their limits using monotonicity and boundedness
5. Use the Monotonic Sequence Theorem to prove convergence without finding the exact limit

Think of this section as building your “sequence toolkit”—a collection of techniques and theorems that you’ll use throughout the rest of calculus and beyond.

Core Concepts

What is a Sequence?

A sequence is a function whose domain is the positive integers. We typically write a sequence as:

where $a_n$ represents the $n$-th term. For example:

• $a_n = \frac{1}{2^n}$ gives us the sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, …$
• $a_n = \frac{n}{n+1}$ gives us $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, …$
• $a_n = (-1)^n\frac{n+1}{3^n}$ gives us $-\frac{2}{3}, \frac{3}{9}, -\frac{4}{27}, \frac{5}{81}, …$

Key Insight: Think of a sequence as an infinite list where each position has exactly one value. The index $n$ tells us which term we’re looking at.

Convergence: Where Does the Sequence Go?

The central question for any sequence is: Does it settle down to a specific value, or does it wander forever?

We say that a sequence ${a_n}$ converges to a limit $L$ if the terms get arbitrarily close to $L$ as $n$ increases. More precisely:

means that for every $\varepsilon > 0$ (no matter how small), there exists some integer $N$ such that whenever $n > N$, we have $\vert a_n - L\vert < \varepsilon$.

In plain English: The terms of the sequence eventually stay within any tiny distance from $L$ that you choose.

Example: The sequence $a_n = \frac{1}{n}$ converges to 0, because the terms $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, …$ get closer and closer to 0.

Counterexample: The sequence $b_n = (-1)^n$ diverges because it oscillates between 1 and -1 forever, never settling down to a single value.

Techniques for Evaluating Limits

When you need to find $\lim_{n\to\infty} a_n$, here are your main strategies:

1. Algebraic Manipulation

For rational sequences, divide numerator and denominator by the highest power of $n$:

2. Known Limits

Memorize these fundamental limits:

• $\lim_{n\to\infty} \frac{1}{n^r} = 0$ for any $r > 0$
• $\lim_{n\to\infty} r^n = 0$ if $\vert r\vert < 1$
• $\lim_{n\to\infty} r^n = 1$ if $r = 1$
• $\lim_{n\to\infty} r^n = \infty$ if $r > 1$

3. Squeeze Theorem

If you can trap your sequence between two simpler sequences with the same limit, you’ve found your answer:

Squeeze Theorem: If $a_n \leq b_n \leq c_n$ for all $n \geq n_0$, and $\lim_{n\to\infty} a_n = \lim_{n\to\infty} c_n = L$, then $\lim_{n\to\infty} b_n = L$.

When to use it: This is powerful when $b_n$ involves oscillating or complicated expressions that are bounded above and below.

4. Connection to Continuous Functions

If $a_n = f(n)$ where $f$ is a continuous function, then:

This means you can use L’Hôpital’s Rule on the continuous version! For example:

Recursive Sequences: Building Each Term from the Last

Some sequences are defined recursively, where each term depends on previous terms. The classic example is the Fibonacci sequence:

To analyze recursive sequences:

1. Compute several terms to understand the pattern
2. Prove monotonicity (is it always increasing? always decreasing?)
3. Find bounds (is it bounded above? below?)
4. Assume convergence and solve for the limit

Example: Consider $a_1 = 2$ and $a_{n+1} = \frac{1}{2}(a_n + 6)$.

• Computing terms: $a_1 = 2, a_2 = 4, a_3 = 5, a_4 = 5.5, a_5 = 5.75, …$
• The sequence appears to be increasing toward 6
• Assuming $\lim_{n\to\infty} a_n = L$, we take limits in the recursion:

Monotonicity and Boundedness: The Power Theorem

One of the most important results in sequence theory:

Monotonic Sequence Theorem: Every bounded monotonic sequence converges.

This is remarkable! Even if you can’t find the exact limit, you can prove convergence exists just by showing:

• The sequence is monotonic (always increasing OR always decreasing), AND
• The sequence is bounded (stays within some finite range)

Definitions:

• Increasing: $a_{n+1} \geq a_n$ for all $n$
• Decreasing: $a_{n+1} \leq a_n$ for all $n$
• Bounded above: There exists $M$ such that $a_n \leq M$ for all $n$
• Bounded below: There exists $m$ such that $a_n \geq m$ for all $n$

Proving monotonicity: Either show $a_{n+1} - a_n \geq 0$ (for increasing), or use induction, or consider $f’(x)$ if $a_n = f(n)$.

Worked Examples

Example 1: Finding the Limit of a Rational Sequence

Problem: Evaluate $\displaystyle \lim_{n\to\infty}\frac{n}{n+1}$.

Thought Process: This is a rational function of $n$. The standard technique is to divide numerator and denominator by the highest power of $n$, which is $n^1$ here.

Solution: \(\frac{n}{n+1} = \frac{n}{n+1} \cdot \frac{\frac{1}{n}}{\frac{1}{n}} = \frac{1}{1+\frac{1}{n}}\)

Now take the limit: \(\lim_{n\to\infty}\frac{1}{1+\frac{1}{n}} = \frac{1}{1+0} = 1\)

Key Takeaway: Dividing by the highest power reveals the dominant behavior. As $n \to \infty$, the $\frac{1}{n}$ term vanishes, leaving just the leading coefficients.

Example 2: Limit Involving a Logarithm

Problem: Evaluate $\displaystyle \lim_{n\to\infty}\frac{\ln n}{n}$.

Thought Process: Logarithms grow very slowly compared to polynomial functions. We expect this limit to be 0. To verify rigorously, we can treat this as a continuous function and apply L’Hôpital’s Rule.

Solution: Let $f(x) = \frac{\ln x}{x}$. As $x \to \infty$, both numerator and denominator approach infinity, so we have the indeterminate form $\frac{\infty}{\infty}$.

Applying L’Hôpital’s Rule: \(\lim_{x\to\infty}\frac{\ln x}{x} = \lim_{x\to\infty}\frac{\frac{d}{dx}[\ln x]}{\frac{d}{dx}[x]} = \lim_{x\to\infty}\frac{1/x}{1} = \lim_{x\to\infty}\frac{1}{x} = 0\)

Therefore, $\lim_{n\to\infty}\frac{\ln n}{n} = 0$.

Key Takeaway: This example demonstrates relative growth rates. Logarithms always lose to polynomials, polynomials lose to exponentials. Understanding these hierarchies helps you quickly estimate limits.

Example 3: Recursive Sequence Convergence

Problem: Consider the sequence defined by: \(a_1 = 2, \quad a_{n+1} = \frac{1}{2}(a_n + 6)\)

Show that ${a_n}$ converges and find its limit.

Thought Process: We’ll use the Monotonic Sequence Theorem. We need to show:

1. The sequence is monotonic (increasing)
2. The sequence is bounded above
3. Then find the limit by solving the fixed-point equation

Solution:

Step 1: Compute several terms to build intuition:

• $a_1 = 2$
• $a_2 = \frac{1}{2}(2 + 6) = 4$
• $a_3 = \frac{1}{2}(4 + 6) = 5$
• $a_4 = \frac{1}{2}(5 + 6) = 5.5$
• $a_5 = \frac{1}{2}(5.5 + 6) = 5.75$

The sequence appears to be increasing toward 6.

Step 2: Prove the sequence is increasing by induction.

• Base case: $a_2 = 4 > 2 = a_1$ ✓
• Inductive step: Assume $a_n > a_{n-1}$. We need to show $a_{n+1} > a_n$.

Since we’ll show $a_n < 6$, this difference is positive, confirming $a_{n+1} > a_n$.

Step 3: Prove the sequence is bounded above by 6 by induction.

• Base case: $a_1 = 2 < 6$ ✓
• Inductive step: Assume $a_n < 6$. Then:

So the sequence is bounded above by 6.

Step 4: Find the limit. By the Monotonic Sequence Theorem, the sequence converges. Let $L = \lim_{n\to\infty} a_n$.

Taking limits on both sides of the recursion: \(L = \lim_{n\to\infty} a_{n+1} = \lim_{n\to\infty} \frac{1}{2}(a_n + 6) = \frac{1}{2}(L + 6)\)

Solving for $L$: \(2L = L + 6 \implies L = 6\)

Key Takeaway: Recursive sequences require a multi-step approach: verify monotonicity and boundedness to ensure convergence, then use the fixed-point equation to find the limit.

Practice Problems

Try these problems to solidify your understanding:

1. Calculate the first five terms and determine the limit (if it exists) of: \(a_n = \frac{(-1)^{n-1}(n+2)}{5^n}\)

2. Does the sequence $b_n = (-1)^n$ converge? Explain why or why not.

3. Evaluate $\displaystyle \lim_{n\to\infty} r^n$ for:
   • (a) $r = \frac{1}{2}$
   • (b) $r = 1$
   • (c) $r = -\frac{1}{2}$
   • (d) $r = 2$
4. Determine whether $c_n = \frac{n}{\sqrt{10+n}}$ converges or diverges.

Hints:

• For problem 1: Notice the exponential denominator dominates
• For problem 2: Can an oscillating sequence have a single limit?
• For problem 3: Think about the cases $\vert r\vert < 1$, $\vert r\vert = 1$, and $\vert r\vert > 1$
• For problem 4: Divide numerator and denominator by $\sqrt{n}$

Key Reminders

As you work through sequences, keep these strategies in mind:

✓ Always check convergence first using the $\varepsilon$-$N$ definition if needed ✓ Use counterexamples like ${(-1)^n}$ to test your understanding of divergence ✓ Compare growth rates (polynomial vs. exponential vs. logarithmic) when evaluating limits ✓ For recursive sequences, prove monotonicity and boundedness before finding the limit ✓ Draw pictures of the first several terms to visualize the sequence’s behavior

Why This Matters

Sequences are everywhere in mathematics and its applications:

• Numerical Analysis: Iterative methods for solving equations generate sequences of approximations
• Computer Science: Algorithms often analyze sequences of states or values
• Physics: Discrete time models use sequences to track system evolution
• Finance: Compound interest and investment growth follow sequence patterns

Mastering sequences now will pay dividends (pun intended!) throughout your mathematical journey.