The Core Distinction: List vs. Sum
A sequence is an ordered list of numbers, often following a specific rule or pattern. A series, in contrast, is the sum of the terms in a sequence. Simply put, a sequence is the list itself, while a series is the result of adding the items in that list together.
Section 2: Notation and Representation
Sequences are typically written as a list of terms separated by commas, enclosed in braces or parentheses, such as {3, 6, 9, 12, ...}. Each number in the list is a 'term'. A series is written as an addition problem, like 3 + 6 + 9 + 12 + ... The focus of a series is on the cumulative total.
Section 3: A Practical Example
Consider the simple sequence of the first four positive odd numbers: {1, 3, 5, 7}. This is the sequence. The corresponding series is the sum of these terms: 1 + 3 + 5 + 7. The value, or sum, of this finite series is 16.
Section 4: Importance in Mathematics
This distinction is critical in higher mathematics, particularly in calculus for studying convergence and divergence, and in finance for calculating things like compound interest. Sequences help model ordered patterns, while series are used to find the total accumulation or a limiting value.