Infinite Sequences

Definition

An infinite sequence is an ordered, unending list of numbers $a_1, a_2, a_3, \dots, a_n, \dots$ , where each term $a_n$ is a value of a function $f(n)$ whose domain is the set of positive integers. It is a discrete mapping from integers to real (or complex) numbers.

Why It Matters

Sequences are the primary tool for modeling processes that unfold over time in discrete steps, like the growth of a population or the iterations of a computer algorithm. They allow us to ask the ultimate long-term question: “Where is this going?” By understanding the limit of a sequence, we can predict the final state of a system without having to simulate every single step of its infinite journey.

Core Concepts

Convergence: A sequence $\{a_n\}$ ${a_{n}}$ converges to a limit $L$ $L$ ( $\lim_{n \to \infty} a_n = L$ $lim_{n \to \infty} a_{n} = L$ ) if the terms become arbitrarily close to $L$ $L$ as $n$ $n$ increases. Formally, for every $\epsilon > 0$ $ϵ > 0$ , there exists $N$ $N$ such that $|a_n - L| < \epsilon$ $∣ a_{n} - L ∣ < ϵ$ for all $n > N$ $n > N$ .
- How to read: “The limit as n approaches infinity of a n is equal to L; for every epsilon greater than zero, there exists N such that the absolute value of the quantity a n minus L is less than epsilon for all n greater than N.”
- Meaning: The tail of the sequence gets trapped in any $\epsilon$ -neighborhood of $L$ — the formal $\epsilon$ - $N$ definition of convergence.
Monotonicity: A sequence is monotonic if it is either non-increasing or non-decreasing.
Boundedness: A sequence is bounded if there exist numbers $M$ and $m$ such that $m \leq a_n \leq M$ for all $n$ .
Sandwich Theorem: If $a_n \leq b_n \leq c_n$ $a_{n} \leq b_{n} \leq c_{n}$ and both $\{a_n\}$ ${a_{n}}$ and $\{c_n\}$ ${c_{n}}$ converge to $L$ $L$ , then $\{b_n\}$ ${b_{n}}$ must also converge to $L$ $L$ .
- How to read: “If a n is less than or equal to b n, which is less than or equal to c n, and both a n and c n converge to L, then b n also converges to L.”
- Meaning: A sequence squeezed between two convergent sequences must converge to the same limit.

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes