1) Introduction: The World of Sequences 📚
Sequences are ordered lists of numbers that often exhibit patterns, and their limits describe their behavior as the number of terms grows infinitely large. This topic introduces sequences and their limits, bridging single-variable calculus to infinite series. Sequences appear in mathematics, computer science, and nature (e.g., Fibonacci numbers), and understanding their limits is key to convergence analysis.
We’ll cover:
- Definition of Sequences: What sequences are and how to denote them.
- Limits of Sequences: Determining if a sequence converges or diverges.
- Limit Laws: Rules to simplify limit calculations.
- Special Sequences: Geometric, arithmetic, and harmonic sequences.
Quick Recap: Limits of functions at infinity prepared us; now we apply this to sequences.
2) Definition of Sequences 🎓
A sequence is a function whose domain is the set of positive integers \( \{1, 2, 3, \ldots\} \), producing a list \( a_1, a_2, a_3, \ldots \). It can be finite or infinite.
Definition 23.1: Sequence
A sequence \( \{a_n\} \) is an ordered list where \( a_n \) is the \( n \)-th term, defined by a formula or rule (e.g., \( a_n = \frac{1}{n} \)).
Example: \( a_n = 2n \) gives \( 2, 4, 6, \ldots \).
Sequences can be explicit (e.g., \( a_n = n^2 \)) or recursive (e.g., \( a_1 = 1 \), \( a_{n+1} = a_n + 2 \)).
Example 1: Listing a Sequence
List the first five terms of \( a_n = \frac{(-1)^n}{n} \).
- \( a_1 = \frac{(-1)^1}{1} = -1 \)
- \( a_2 = \frac{(-1)^2}{2} = \frac{1}{2} \)
- \( a_3 = \frac{(-1)^3}{3} = -\frac{1}{3} \)
- \( a_4 = \frac{(-1)^4}{4} = \frac{1}{4} \)
- \( a_5 = \frac{(-1)^5}{5} = -\frac{1}{5} \)
Answer: \( -1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, -\frac{1}{5} \).
Example 2: Recursive Sequence
Find the first five terms of \( a_1 = 1 \), \( a_{n+1} = 2a_n + 1 \).
- \( a_1 = 1 \)
- \( a_2 = 2 \cdot 1 + 1 = 3 \)
- \( a_3 = 2 \cdot 3 + 1 = 7 \)
- \( a_4 = 2 \cdot 7 + 1 = 15 \)
- \( a_5 = 2 \cdot 15 + 1 = 31 \)
Answer: \( 1, 3, 7, 15, 31 \).
3) Limits of Sequences 📐
The limit of a sequence \( \{a_n\} \) as \( n \to \infty \) describes its long-term behavior. A sequence converges to \( L \) if \( a_n \) gets arbitrarily close to \( L \).
Definition 23.2: Limit of a Sequence
\( \lim_{n \to \infty} a_n = L \) if for every \( \epsilon > 0 \), there exists \( N \) such that \( |a_n - L| < \epsilon \) for all \( n > N \). If no such \( L \) exists, the sequence diverges.
**Intuition**: As \( n \) increases, \( a_n \) approaches \( L \) or goes to \( \infty \), \( -\infty \), or oscillates.
Example 3: Convergent Sequence
Find \( \lim_{n \to \infty} \frac{3n + 1}{n} \).
- Divide by \( n \): \( \frac{3n + 1}{n} = 3 + \frac{1}{n} \).
- As \( n \to \infty \), \( \frac{1}{n} \to 0 \).
- Limit: \( 3 + 0 = 3 \).
Answer: \( 3 \).
Example 4: Divergent Sequence
Determine \( \lim_{n \to \infty} n^2 \).
- As \( n \) increases, \( n^2 \) grows without bound.
- Limit: \( \infty \) (diverges).
Answer: Diverges to \( \infty \).
4) Limit Laws for Sequences 🔍
Limit laws simplify finding limits, similar to function limits, assuming the limits exist.
Definition 23.3: Limit Laws
- Sum: \( \lim (a_n + b_n) = \lim a_n + \lim b_n \). - Product: \( \lim (a_n b_n) = (\lim a_n) (\lim b_n) \). - Quotient: \( \lim \frac{a_n}{b_n} = \frac{\lim a_n}{\lim b_n} \) if \( \lim b_n \neq 0 \). - Constant Multiple: \( \lim c a_n = c \lim a_n \).
Example 5: Using Limit Laws
Find \( \lim_{n \to \infty} \left( \frac{1}{n} + \frac{2}{n^2} \right) \).
- \( \lim \frac{1}{n} = 0 \), \( \lim \frac{2}{n^2} = 0 \).
- Sum: \( 0 + 0 = 0 \).
Answer: \( 0 \).
Example 6: Product Rule
Compute \( \lim_{n \to \infty} \left( \frac{n}{n+1} \cdot \frac{2n}{n-1} \right) \).
- \( \lim \frac{n}{n+1} = \lim \frac{1}{1 + \frac{1}{n}} = 1 \).
- \( \lim \frac{2n}{n-1} = \lim \frac{2}{1 - \frac{1}{n}} = 2 \).
- Product: \( 1 \cdot 2 = 2 \).
Answer: \( 2 \).
5) Special Sequences and Their Limits 🔍
Certain sequences have well-known limits, such as geometric sequences (\( a_n = ar^n \)) and harmonic sequences (\( a_n = \frac{1}{n} \)).
Definition 23.4: Special Sequences
- Geometric: \( a_n = ar^n \), limit is 0 if \( |r| < 1 \), diverges if \( |r| \geq 1 \). - Harmonic: \( a_n = \frac{1}{n} \), limit is 0.
Example 7: Geometric Sequence
Find \( \lim_{n \to \infty} 3 \cdot \left(\frac{1}{2}\right)^n \).
- Form: \( a_n = 3 \cdot \left(\frac{1}{2}\right)^n \), \( r = \frac{1}{2} < 1 \).
- Limit: \( 3 \cdot 0 = 0 \).
Answer: \( 0 \).
Example 8: Harmonic Sequence
Determine \( \lim_{n \to \infty} \frac{1}{n^2} \).
- As \( n \to \infty \), \( n^2 \to \infty \), so \( \frac{1}{n^2} \to 0 \).
Answer: \( 0 \).
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Find the limit of the given sequence, or determine if it diverges. 📚
\( a_n = \frac{2n}{n+1} \)
\( a_n = \frac{1}{n^2} \)
\( a_n = (-1)^n \)
\( a_n = 3^n \)
\( a_n = \frac{n+2}{2n-1} \)
\( a_n = \frac{1}{2^n} \)
\( a_n = n \sin\left(\frac{1}{n}\right) \)
\( a_n = \frac{2n^2 + 1}{n^2 + n} \)
\( a_n = \cos\left(\frac{\pi n}{2}\right) \)
\( a_n = \frac{n}{n^2 + 1} \)
\( a_n = 5 \cdot \left(\frac{1}{3}\right)^n \)
Challenging Practice Questions 🌟
Instructions: Analyze the limit of these sequences with advanced techniques or interpretations. 🧠
Find \( \lim_{n \to \infty} \frac{n^2 + n}{n^2 - n} \) and explain its behavior.
Determine if \( a_n = \frac{\sin(n)}{n} \) converges or diverges, and justify.
Compute \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \) and relate to a known constant.
Evaluate \( \lim_{n \to \infty} \frac{n!}{n^n} \) and discuss its convergence.
Analyze \( a_n = \frac{(-1)^n n}{n+1} \) for limit and oscillatory behavior.
7) Summary & Cheat Sheet 📋
7.1) Sequence
\( \{a_n\} \) is a list of terms defined by a rule or recursion.
7.2) Limit
\( \lim_{n \to \infty} a_n = L \) if \( a_n \) approaches \( L \); diverges otherwise.
7.3) Limit Laws
Use sum, product, quotient, and constant multiple rules for convergence.
You’ve mastered sequences and their limits! Next, we’ll explore infinite series. 🎉