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A sequence is an ordered list of numbers, where each number is determined by a specific rule or formula. Each element in a sequence is usually denoted as a_n, where n is the position of the element in the sequence. For example, the sequence {2, 4, 6, 8, ...} follows the rule 2n, with a_1 = 2, a_2 = 4, a_3 = 6, and so on.

A function is a relation between two sets that associates each element in the first set (the domain) with exactly one element in the second set (the range), according to a specific rule. A function can be represented as f(x), where x belongs to the domain and f(x) is the corresponding output in the range.

In the context of an infinite sequence, we can consider f(n) = a_n, where n is a positive integer. This means that the function f takes a positive integer as its input (n) and relates it to the corresponding element in the infinite sequence (a_n). This holds true for all positive integers. In other words, the domain of the function consists of the positive integers while the range is determined by the elements of the sequence.

Representing an infinite sequence as a function allows us to utilize the powerful language and tools of functions to analyze the sequence. For example, we can investigate the properties of continuous functions, find limits, calculate derivatives and integrals, and solve differential equations, all of which apply to sequences when viewed as functions that map positive integers to the sequence elements. In summary, an infinite sequence can be defined as a function with its domain being the set of positive integers because there is a clear relationship between the elements of the sequence and the positive integers, commonly represented as f(n) = a_n for all n belonging to the domain. This representation allows us to better understand, analyze, and work with the properties of sequences using the tools of functions.