Sequences and Infinite Series

One of the most vital characteristics of any sequence is understanding how its terms behave as the index variable $n$ grows infinitely large. When we evaluate a sequence's ultimate destination, we are analyzing its mathematical limit.

Conceptual Overview

When tracking a sequence out toward infinity ($\infty$), its terms will generally exhibit one of three behaviors:

1. They will approach a single, fixed numerical value.

2. They will grow infinitely large in magnitude without any bound.

3. They will oscillate back and forth erratically with no clear pattern.

Based on these behaviors, we mathematically classify sequences into two distinct categories:

While a sequence is just a list of numbers, an infinite series is the result of adding that infinite list of numbers together. Since you cannot physically perform an infinite number of additions, calculus provides frameworks to determine if an endless sum adds up to a clean, finite number or explodes to infinity.

Conceptual Overview

An infinite series is written using summation notation:

To evaluate whether this endless addition settles on a finite total, we analyze its structural behavior using two primary approaches:

A sequence is simply an ordered list of numbers governed by a specific mathematical rule. In engineering and calculus, we represent these lists using index variables (usually $n$) to denote the position of each number in the line.

There are two primary ways to define and generate these sequences: implicitly (via a recurrence relation) and explicitly (via an explicit formula).

Conceptual Overview