Technical Mathematics, Sixth Edition

20.1. Sequences and Series

Before using sequences and series, we must define some terms.

20.1.1. Sequences

A sequence

i₁, u₂, u₃, ..., u_n

is a set of quantities, called terms, which follow each other in a definite order. Each term can be determined either by its position in the sequence or by knowledge of the preceding terms.

Example 1: Here are some different kinds of sequences. The sequence 1, , ..., 1/n is called a finite sequence because it has a finite number of terms, n. The sequence 1, , ..., 1/n, ... is called an infinite sequence. The three dots at the end indicate that the sequence continues indefinitely. The sequence 3, 7, 11, 15, ... is called an arithmetic sequence, or arithmetic progression (AP), because each term after the first is equal to the sum of the preceding term and a constant. That constant (4, in this example) is called the common difference. The sequence 2, 6, 18, 54, ... is called a geometric sequence, or geometric progression (GP), because each term after the first is equal to the product of the preceding term and a constant. That constant (3, in this example) is called the common ratio. The sequence 1, 1, 2, 3, 5, 8, ... is called a Fibonacci sequence. Each term after the first two terms is the sum of the two preceding terms.

20.1.2. Series

A series is ...