Sequences for MATH 141

What is a Sequence?

A sequence is simply a list of numbers in a specific order. Unlike a set, order matters and repetition is allowed.

$$1, 4, 9, 16, 25, \ldots$$

This is the sequence of perfect squares. The "..." means the pattern continues forever.

Sequence Notation

There are several ways to write sequences:

1. List notation: $1, 4, 9, 16, 25, \ldots$

2. Set-builder notation: $\{a_n\}_{n=1}^{\infty}$ or simply $\{a_n\}$

3. Formula notation: $a_n = n^2$ for $n = 1, 2, 3, \ldots$

The Subscript $n$

The subscript $n$ is called the index. It tells you which term you're looking at:

• $a_1$ = first term
• $a_2$ = second term
• $a_n$ = the $n$th term (general term)

Example: If $a_n = n^2$, then:

• $a_1 = 1^2 = 1$
• $a_2 = 2^2 = 4$
• $a_3 = 3^2 = 9$
• $a_{10} = 10^2 = 100$

Starting Index

Sequences don't always start at $n = 1$. Watch the notation!

$$\{a_n\}_{n=0}^{\infty} \text{ starts at } n = 0$$

$$\{a_n\}_{n=3}^{\infty} \text{ starts at } n = 3$$

Finding the Formula from a Pattern

Limits of Sequences

The big question: What happens to $a_n$ as $n \to \infty$?

• If $a_n$ approaches a finite number $L$, the sequence converges to $L$
• If $a_n$ doesn't approach a finite number, the sequence diverges

$$\lim_{n \to \infty} a_n = L \quad \text{means the sequence converges to } L$$

Useful Limit Facts

For sequences, these limits are essential:

Recursive Sequences

Some sequences are defined recursively — each term depends on previous terms.

Example: The Fibonacci sequence

$$a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2} \text{ for } n \geq 3$$

This gives: $1, 1, 2, 3, 5, 8, 13, 21, \ldots$

Bounded and Monotonic Sequences

Bounded: A sequence is bounded if all terms stay within some fixed range.

• Bounded above: There's a number $M$ where $a_n \leq M$ for all $n$
• Bounded below: There's a number $m$ where $a_n \geq m$ for all $n$

Monotonic: A sequence is monotonic if it only goes one direction.

• Increasing: $a_1 < a_2 < a_3 < \cdots$ (or $\leq$)
• Decreasing: $a_1 > a_2 > a_3 > \cdots$ (or $\geq$)

Key Theorem: A bounded monotonic sequence always converges.

Common Mistakes and Misunderstandings

❌ Mistake: Confusing $a_n$ with $a(n)$

Wrong: Treating $a_n$ as function notation $a(n)$ and writing $a_n = a \cdot n$.

Why it's wrong: The subscript $n$ is an index, not multiplication. $a_n$ means "the $n$th term of sequence $a$."

Correct: $a_n$ is read as "a sub n" — it's the term at position $n$.

❌ Mistake: Ignoring the starting index

Wrong: Assuming every sequence starts at $n = 1$.

Why it's wrong: $\{a_n\}_{n=0}^{\infty}$ has first term $a_0$, not $a_1$. The starting index changes which term is "first."

Correct: Always check the starting index. For $a_n = 2^n$ starting at $n = 0$: first term is $a_0 = 2^0 = 1$.

❌ Mistake: Saying a sequence "diverges to -1 and 1"

Wrong: "$(-1)^n$ diverges to both -1 and 1."

Why it's wrong: Divergence doesn't mean "going to multiple values." The sequence diverges because it fails to approach ANY single value.

Correct: "$(-1)^n$ diverges because it oscillates — it doesn't approach a limit."

❌ Mistake: Thinking polynomial can beat exponential for large n

Wrong: "$n^{1000}$ is huge, so $\frac{n^{1000}}{2^n} \to \infty$."

Why it's wrong: For large enough $n$, exponential growth ALWAYS overtakes polynomial growth. Always. No matter how large the exponent.

Correct: $\displaystyle\lim_{n \to \infty} \frac{n^{1000}}{2^n} = 0$. Exponential wins.