**Sequences and Series**

A. OBJ: to find the sum of a series, to use sequence notation, to use factorial notation and to use series notation.

B. FACTS/FORMULAS:

1. sequence means that it is ordered so that it has a first member, a second member, a third member, and so on. Mathematically, you can think of a sequence as afunction whose domain is the set of positive integers.

2. sequences are usually written using subscript notation

*a*<sub>1</sub>, *a*<sub>2</sub>, *a*<sub>3</sub>, … .

3.

**Definition of Sequence**

An **infinite sequence** is a function whose domain is the set of positive integers.

The function values

*a*<sub>1</sub>, *a*<sub>2</sub>, *a*<sub>3</sub>, *a*<sub>4</sub>, …, *a<sub>n</sub>*, … .

are the **terms** of the sequence. When the domain of the function consists of the first n positive integers only. the sequence is a **finite sequence**.

A sequence is called **finite sequence** if it has finite terms e.g., 2, 4, 6, 8, 10, 12, 14, 16.

A sequence is called **infinite sequence** if it has infinite terms, e.g., 4, 6, 8, 10, 12, 14, …

*Overview*

By a sequence, we mean an arrangement of numbers in a definite order according to some rule. We denote the terms of a sequence by *a*<sub>1</sub>, *a*<sub>2</sub>, *a*<sub>3</sub>, …, etc., the subscript denotes the position of the term.

A sequence is either finite or infinite depending upon the number of terms in a sequence. We should not expect that its terms will be necessarily given by a specific formula.

However, we expect a theoretical scheme or rule for generating the terms.

Let *a*<sub>1</sub>, *a*<sub>2</sub>, *a*<sub>3</sub>, …, be the sequence, then, the expression *a*<sub>1</sub>+*a*<sub>2</sub>+*a*<sub>3</sub>+⋯ is called the **series** associated with given sequence. The series is finite or infinite according as the given sequence is finite or infinite.

*Remark*: When the series is used, it refers to the indicated sum not to the sum itself. Sequence following certain patterns are more often called **progressions**. In rogressions, we note that each term except the first progresses in a definite manner.

**Progression**:

If a sequence of number is such that each term can be obtained from the preceding one by the operation of some law, the sequence is called a progression.

*Note*:- Each progression is a sequence but each sequence may or may not be a progression.

read more: **What is a Sequence**?