Sequence

A sequence is an ordered list of terms that often follow a specific pattern or rule. Each term in the sequence has a unique position denoted by a positive integer, usually denoted by the variable .

For example, the arithmetic sequence has a rule that each term is two more than the previous term. The rule for a general term in this sequence can be represented as where is the position of the term in the sequence.

A sequence can be finite, having a specific number of terms, or infinite, continuing indefinitely. The terms of a sequence can be generated using a formula or a recursive rule. In calculus, sequences are used to study the behavior of functions, convergence, and limits.

Series

A series is the sum of the terms of a sequence. It can be finite or infinite, depending on the underlying sequence. In calculus, series are used to approximate functions, study convergence and divergence, and understand the behavior of infinite sums.