Key Concepts of Sequences

Why This Matters

Sequences are the foundation of everything you'll encounter in Calculus II—from infinite series to Taylor polynomials to convergence tests. When you're asked whether a series converges, you're really being asked about the behavior of its underlying sequence. The concepts here—convergence, boundedness, monotonicity, and limit evaluation—show up repeatedly in exam questions, often disguised as series problems or applications of specific convergence tests.

Don't just memorize definitions here. You're being tested on your ability to identify sequence behavior, apply convergence criteria, and connect sequences to series. Every concept in this guide links directly to tools you'll use later, so focus on understanding why a sequence converges or diverges, not just whether it does. Master these fundamentals now, and series convergence tests will feel like natural extensions rather than new material.

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Foundational Definitions

Before analyzing sequence behavior, you need to understand what sequences are and how we describe them mathematically. A sequence is simply a function from the natural numbers to the real numbers, producing an ordered list of outputs.

Definition of a Sequence

• Ordered list of numbers—each term $a_n$ corresponds to a natural number index $n$, creating a function $a: \mathbb{N} \to \mathbb{R}$
• Terms are indexed starting at $n = 1$ (or sometimes $n = 0$)—the choice of starting index affects formulas but not convergence behavior
• Infinite sequences are the focus in Calculus II—finite sequences rarely appear on exams since convergence only matters when $n \to \infty$

Limit of a Sequence

• The limit $L$ is the value terms approach—formally, $\lim_{n \to \infty} a_n = L$ means for any $\epsilon > 0$, there exists $N$ such that $|a_n - L| < \epsilon$ for all $n > N$
• Algebraic manipulation is your primary tool—factor, rationalize, or divide by the highest power of $n$ to evaluate limits
• L'Hôpital's Rule applies when you can write $a_n = f(n)$ for a differentiable function $f(x)$ with indeterminate form

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Convergence Criteria

Understanding when and why sequences converge is the core skill tested in this unit. Convergence means the terms settle toward a single finite value; divergence means they don't.

Convergence and Divergence of Sequences

• Convergent sequences approach a finite limit $L$—the terms eventually stay arbitrarily close to $L$ and never "escape"
• Divergent sequences either grow without bound ($\to \pm\infty$), oscillate, or fail to settle—oscillation is a common exam trap
• Convergence of $a_n$ to $L \neq 0$ means $\sum a_n$ diverges by the Divergence Test—this connection appears constantly

Monotonic Sequences

• Monotonically increasing means $a_{n+1} \geq a_n$ for all $n$; monotonically decreasing means $a_{n+1} \leq a_n$
• Test monotonicity by examining $a_{n+1} - a_n$ (positive = increasing) or $\frac{a_{n+1}}{a_n}$ (greater than 1 = increasing for positive terms)
• Monotonic + bounded = convergent—this is the Monotone Convergence Theorem, a powerful tool when direct limit evaluation fails

Bounded Sequences

• Bounded above means there exists $M$ with $a_n \leq M$ for all $n$; bounded below means $a_n \geq m$; bounded means both
• Boundedness alone doesn't guarantee convergence—the sequence $a_n = (-1)^n$ is bounded but diverges by oscillation
• Combined with monotonicity, boundedness ensures convergence—this pairing is essential for proving convergence without finding the explicit limit

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Special Sequence Types

Certain sequence families appear so frequently that you should recognize their forms and convergence behavior instantly. Knowing the general term formula lets you quickly determine limits and behavior.

Arithmetic Sequences

• Constant difference $d$ between terms—the general term is $a_n = a_1 + (n-1)d$, producing a linear pattern
• Always diverge (unless $d = 0$)—since $a_n \to \pm\infty$ as $n \to \infty$ when $d \neq 0$
• Sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ applies to partial sums—useful for finite sums but confirms divergence for series

Geometric Sequences

• Constant ratio $r$ between terms—the general term is $a_n = a_1 \cdot r^{n-1}$
• Convergence depends entirely on $|r|$—converges to 0 if $|r| < 1$, diverges if $|r| \geq 1$ (oscillates when $r \leq -1$)
• Foundation for geometric series—the series $\sum a_1 r^{n-1}$ converges to $\frac{a_1}{1-r}$ only when $|r| < 1$

Recursive Sequences

• Defined by a recurrence relation—each term depends on previous terms, like $a_{n+1} = f(a_n)$ or $a_n = a_{n-1} + a_{n-2}$
• Fibonacci sequence is the classic example—$F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$
• Finding limits often requires assuming convergence: if $\lim a_n = L$ exists, substitute $L$ into the recurrence and solve

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Key Theorems and Techniques

These tools help you prove convergence when direct computation isn't possible. Mastering these techniques separates students who can handle tricky exam problems from those who can't.

Squeeze Theorem for Sequences

• If $b_n \leq a_n \leq c_n$ and $\lim b_n = \lim c_n = L$, then $\lim a_n = L$—the sequence is "squeezed" to the same limit
• Ideal for oscillating factors—sequences like $a_n = \frac{\sin n}{n}$ are bounded by $\pm\frac{1}{n}$, both converging to 0
• Requires finding bounding sequences—look for terms you can replace with their maximum/minimum values

Infinite Series and Their Relationship to Sequences

• A series $\sum_{n=1}^{\infty} a_n$ is the limit of partial sums $S_N = \sum_{n=1}^{N} a_n$—series convergence means the sequence $\{S_N\}$ converges
• If $\sum a_n$ converges, then $\lim a_n = 0$—but the converse is false (harmonic series is the classic counterexample)
• Sequence behavior informs series tests—the Ratio Test, Root Test, and Comparison Tests all analyze sequences to determine series convergence

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Quick Reference Table

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Self-Check Questions

1. A sequence is bounded and monotonically decreasing. What can you conclude about its convergence, and which theorem guarantees this?

2. Compare the long-term behavior of the arithmetic sequence $a_n = 3 + 2n$ and the geometric sequence $b_n = 3 \cdot (0.5)^n$. Which converges, and why?

3. Given $a_n = \frac{n \cos n}{n^2 + 1}$, which theorem would you use to find $\lim_{n \to \infty} a_n$, and what bounding sequences would you choose?

4. If $\lim_{n \to \infty} a_n = 5$, what does this tell you about the series $\sum_{n=1}^{\infty} a_n$? Explain using the Divergence Test.

5. A recursive sequence is defined by $a_1 = 2$ and $a_{n+1} = \frac{1}{2}(a_n + 6)$. Assuming the sequence converges, find its limit. What additional properties would you need to verify to prove convergence?