Sequences and Series On occasion, it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become

Simply listing the first few terms is not sufficient to define a unique sequence-----the nth term must be given. Although the first three terms are the same, these are different sequences We can only write an apparent nth term. There may be others

apparent pattern:

Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. Write the first five terms of this sequence. The subscripts of the sequence make up the domain of the sequence and they identify the location of a term within the sequence.

Factorial Notation zero factorial is defined as 0! = 1 Factorials follow the same rules for order of operations as exponents. 2n! = 2(n!) =

There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase Greek letter sigma.

Properties of Sums

Series Many applications involve the sum of the terms of a finite or an infinite sequence. Such a sum is called a series.

Notice that the sum of an infinite series can be a finite number. Variations in the upper and lower limits of summation can produce quite different-looking summation notation for the same sum.

Sequences have many applications in situations that involve a recognizable pattern.