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.
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>, … .
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, …
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.
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?