Sequences and Series

Sequences

A sequence is an ordered list of numbers. Each number in the sequence is often the result of a rule or pattern. For example, the sequence 2, 4, 6, 8, 10,... is an arithmetic sequence where each term after the first is found by adding 2 to the previous term.

Sequences can be finite or infinite. The value of a sequence at a particular term n is often written as a_n or a(n).

Series

A series is the sum of the terms of a sequence. If the sequence is finite, the series is a simple sum of a finite number of terms. If the sequence is infinite, the series is an infinite sum and it's not always clear if the sum is finite or infinite.

A series can be represented in sigma notation as Σ a_n where n runs from 1 to infinity.

If the sum of an infinite series is a finite number, we say the series converges. If the sum of an infinite series is not a finite number, we say the series diverges.

Importance in Mathematics

Sequences and series are fundamental concepts in mathematics and they appear in various areas such as algebra, calculus, number theory, and complex analysis. They are used to define functions, solve equations, and even to define other mathematical structures. In calculus, they are used to define limits, continuity, and differentiability. In number theory, they are used to study the properties of numbers.

Understanding sequences and series is crucial for understanding more advanced concepts in mathematics.