The terms sequence and series are fundamental concepts in mathematics, particularly in the study of numbers and functions. While they may seem similar at first glance, they refer to different ideas.
A sequence is an ordered list of numbers, typically following a specific rule or pattern. For example, the sequence of natural numbers can be expressed as: 1, 2, 3, 4, 5, …. In this case, each number is a term in the sequence, and the sequence continues indefinitely. Sequences can be finite (having a limited number of terms) or infinite (having an unlimited number of terms). They are often defined by a formula or a recursive relation, indicating how to generate subsequent terms from previous ones.
In contrast, a series is the sum of the terms of a sequence. When you take the numbers in a sequence and add them together, you create a series. For example, if we consider the sequence 1, 2, 3, 4, 5, the associated series would be the sum of those terms: 1 + 2 + 3 + 4 + 5 = 15. Just like sequences, series can also be finite or infinite. An example of an infinite series is the sum of all natural numbers: 1 + 2 + 3 + 4 + ….
To summarize, the key difference is that a sequence is a list of numbers, while a series is the total obtained by adding those numbers together. Understanding this distinction is crucial in many areas of mathematics, including calculus and number theory, as it helps in effectively analyzing patterns and their properties.