Sequences Recalled and Reframed
What Is a Sequence, Really?
A sequence is a function from the natural numbers to the real numbers:
$a : \mathbb{N} \to \mathbb{R}, \qquad n \mapsto a_n.$
We write the sequence as $(a_n)_{n=1}^{\infty}$ or simply $(a_n)$. Each natural number $n$ is an index and $a_n$ is the $n$-th term.
Ways to Define a Sequence
| Method | Example |
|---|---|
| Explicit formula | $a_n = \dfrac{1}{n}$ |
| Recurrence relation | $a_1 = 1,\; a_{n+1} = \sqrt{1 + a_n}$ |
| Verbal / piecewise | "$a_n$ is the $n$-th prime number" |
Key Vocabulary
- Monotone increasing: $a_{n+1} \ge a_n$ for all $n$.
- Monotone decreasing: $a_{n+1} \le a_n$ for all $n$.
- Bounded above: $\exists\, M$ such that $a_n \le M$ for all $n$.
- Bounded below: $\exists\, m$ such that $a_n \ge m$ for all $n$.
- Bounded: bounded above and below, i.e. $|a_n| \le K$ for some $K$.
Familiar Friends Revisited
Arithmetic progression: $a_n = a_1 + (n-1)d$. If $d \neq 0$ the terms escape to $\pm\infty$; the sequence is unbounded.
Geometric progression: $a_n = a_1 \cdot r^{n-1}$.
- $|r| < 1$: terms shrink toward $0$.
- $r = 1$: constant sequence.
- $|r| > 1$: terms blow up.
- $r = -1$: oscillates between $\pm a_1$.
These examples already hint at the central question: what happens to $a_n$ as $n \to \infty$?
Viewpoint shift: In earlier algebra you computed individual terms. Now we study the long-term behaviour — does the sequence settle down, oscillate, or escape?
Intuitive Idea of Convergence
Before any formal definition, let us build intuition. A sequence converges to a number $L$ if its terms get and stay arbitrarily close to $L$. "Get" means the terms approach $L$; "stay" means they don't wander away again.