 |
| Sequences and Series |
|
|
| |
| 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 an
or tn. |
| |
| 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 an or tn. |
| |
|
| |
| 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. |
| |
|
| |
| 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. |
| |
| |
