Sequences and Series

Sequence

A set of numbers arranged in a definite order according to some definite rule is called a sequence.

or

A sequence is a function whose domain is the set N of natural numbers.

It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n Î N under 'a' by a_n or t_n.

Examples:

1, 3, 5, 7…..... (adding 2 to every term)

1, 4, 16, 64 … (Multiplying by 4 every term)

20, 17, 14 … . (add -3 to every term)

The different numbers in a sequence are called terms of sequence.

The subscripts denote the position of the term.

In the second example, 4 is the second term, and 14 is the third term in the third example.

The nth term of a sequence is called the general term of the sequence and is usually denoted by a_n or t_n.

Finite and Infinite Sequences

A sequence is called finite if the number of terms is finite. A finite sequence has always a last term.

Examples:

2, 5, 8, 11, 14 …, 32

37, 33 …, 1

A sequence is called infinite if the number of terms is infinite. An infinite sequence has no last term. In this sequence, every term is followed by a new term.

Examples:

i) A sequence of multiples of 5

5, 10, 15, 20, …

ii) A sequence of reciprocals of positive integers

The above two sequences are clearly the infinite sequences.

Series

Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.

Examples:

i) 1 + 4 + 7 + 10 + ... is a series in which first term is 1, second term is 4, third term is 7 and so on.

ii) 3 - 9 + 27 - 81 + ... is also a series in which the first term is 3, second term is -9, third term is 27 and so on.