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.

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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 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 that terms approach as $n$ grows. Formally, $\lim_{n \to \infty} a_n = L$ means that for any $\epsilon > 0$, there exists an integer $N$ such that $|a_n - L| < \epsilon$ for all $n > N$. In plain terms: no matter how tight a band you draw around $L$, the sequence eventually stays inside it permanently.

To actually evaluate limits, you have a few go-to strategies:

• Algebraic manipulation: factor, rationalize, or divide numerator and denominator by the highest power of $n$
• L'Hôpital's Rule: when you can write $a_n = f(n)$ for a differentiable function $f(x)$ that produces an indeterminate form like $\frac{\infty}{\infty}$ or $\frac{0}{0}$, differentiate top and bottom with respect to $x$
• Dominant term analysis: in a ratio of polynomials, the highest-degree terms in the numerator and denominator determine the limit. For example, $\frac{3n^2 + n}{5n^2 - 2} \to \frac{3}{5}$

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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: a sequence like $(-1)^n$ doesn't blow up, but it still diverges because it never settles on one value
• The bridge to series: if $a_n \to L$ and $L \neq 0$, then $\sum a_n$ diverges by the Divergence Test. This connection appears constantly, so internalize it now

Monotonic Sequences

A sequence is monotonically increasing if $a_{n+1} \geq a_n$ for all $n$, and monotonically decreasing if $a_{n+1} \leq a_n$ for all $n$. (Strict monotonicity uses $>$ or $<$ instead.)

Two reliable ways to test monotonicity:

1. Difference test: compute $a_{n+1} - a_n$. If it's always positive, the sequence is increasing. If always negative, decreasing.
2. Ratio test: for sequences with positive terms, compute $\frac{a_{n+1}}{a_n}$. Greater than 1 means increasing; less than 1 means decreasing.

The payoff: Monotonic + bounded = convergent. This is the Monotone Convergence Theorem, and it's a powerful tool when you can't find the limit directly. You don't need to know what the limit is to prove it exists.

Bounded Sequences

• Bounded above means there exists $M$ with $a_n \leq M$ for all $n$; bounded below means $a_n \geq m$ for all $n$; bounded means both
• Boundedness alone doesn't guarantee convergence. The sequence $a_n = (-1)^n$ is bounded between $-1$ and $1$ but diverges by oscillation
• Combined with monotonicity, boundedness ensures convergence. This pairing is especially useful for recursive sequences where you can't easily compute the limit directly

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Special Sequence Types

Certain sequence families appear so frequently that you should recognize their forms and convergence behavior on sight.

Arithmetic Sequences

An arithmetic sequence has a constant difference $d$ between consecutive terms, with general term $a_n = a_1 + (n-1)d$.

These always diverge unless $d = 0$ (a constant sequence), because the terms grow linearly toward $\pm\infty$. The partial sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ is useful for finite sums but confirms that the corresponding series diverges.

Geometric Sequences

A geometric sequence has a constant ratio $r$ between consecutive terms, with general term $a_n = a_1 \cdot r^{n-1}$.

Convergence depends entirely on $|r|$:

• If $|r| < 1$: the sequence converges to 0
• If $|r| = 1$: the sequence is constant ($r = 1$) or oscillates between $\pm a_1$ ($r = -1$)
• If $|r| > 1$: the terms blow up in magnitude

This is also the foundation for geometric series: $\sum_{n=1}^{\infty} a_1 r^{n-1}$ converges to $\frac{a_1}{1-r}$ only when $|r| < 1$.

Recursive Sequences

Recursive sequences are defined by a recurrence relation where each term depends on previous terms, like $a_{n+1} = f(a_n)$. The Fibonacci sequence is the classic example: $F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$.

To find the limit of a recursive sequence (when it converges):

1. Assume the limit $L$ exists
2. Since $a_n \to L$ and $a_{n+1} \to L$, substitute $L$ for both in the recurrence relation
3. Solve the resulting equation for $L$

This technique only finds candidate limits. To prove convergence, you still need to verify that the sequence is monotonic and bounded (or use another convergence argument).

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Key Theorems and Techniques

These tools help you prove convergence when direct computation isn't straightforward.

Squeeze Theorem for Sequences

If you can trap a sequence between two others that share the same limit, the trapped sequence must converge to that limit too. Formally: if $b_n \leq a_n \leq c_n$ for all $n$ beyond some point, and $\lim_{n \to \infty} b_n = \lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} a_n = L$.

This is ideal for oscillating factors. For example, $a_n = \frac{\sin n}{n}$: since $-1 \leq \sin n \leq 1$, you get $\frac{-1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}$. Both bounds converge to 0, so $a_n \to 0$.

The main challenge is finding good bounding sequences. Look for terms you can replace with their maximum or minimum possible values.

Infinite Series and Their Relationship to Sequences

A series $\sum_{n=1}^{\infty} a_n$ is defined as the limit of its partial sums $S_N = \sum_{n=1}^{N} a_n$. So series convergence really means the sequence $\{S_N\}$ converges.

Two critical facts connect sequences to series:

• If $\sum a_n$ converges, then $\lim_{n \to \infty} a_n = 0$. But the converse is false. The harmonic series $\sum \frac{1}{n}$ diverges even though $\frac{1}{n} \to 0$. This is probably the most important counterexample in the course.
• If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges. This is the Divergence Test (also called the $n$th-Term Test). It can only prove divergence, never convergence.

Sequence behavior also drives the Ratio Test, Root Test, and Comparison Tests, all of which 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?