9.1 Sequences and Series
An infinite sequence is an unending “succession” of numbers called terms. The terms have a
definite order.
a1
a2
an
1st term
2nd term
nth term
Examples of sequences:
1, 2,3, 4,...
1,3,5, 7,9,...
1,1, 2,3,5,8,13, 21,...
Fibonacci’s Sequence
Sequences are often written in brace notation.

1,3,5, 7,9,..., 2n  1,... can be written as 2n  1n 1


1 1 1
1
 1 
1, , , ,..., n1 ,... can be written as  n 1 
3 9 27
3
 3 n 1
or
1
 n
 3 n  0
Consider the sequence of odd numbers 1,3,5, 7,9,...2n  1,... .
We can denote the general term by f  n   2n  1 then we have f 1 , f  2  , f  3 ,..., f  n  ...
Definition: A sequence is a function whose domain is a set of integers. Specifically, we will

regard the expression an n 1 to be an alternative notation for the function f  n   an , n  1, 2,3...
If the domain of the function consists of the first n positive integers only, the sequence is a finite
sequence.
Some sequences are defined recursively. To define a sequence recursively, you need to be
given one or more of the first few terms. All of the other terms of the sequence are then defined
using previous terms.
Factorial Notation
If n is a positive integer, n factorial is defined as n!  n  n  1 n  2  ...  3 2 1
0! =1 by definition
Summation Notation: The sum of the first n terms of a sequence is represented by
n
a
i 1
i
 a1  a2  ...  an where i is called the index of summation, n is the upper limit of
summation, and 1 is the lower limit of summation
Properties of Sums
n
1.
 c  cn
i 1
2.
3.
n
n
i 1
i 1
 cai  c ai
n
n
n
i 1
i 1
i 1
  ai  bi    ai   bi
Definition: Given a sequence of numbers a1 , a2 , a3 , a4 ,... the sum of the first n terms of the
sequence is called the nth partial sum.
S1  a1
S2  a1  a2
S3  a1  a2  a3
.
.
.
n
Sn  a1  a2  ...  an   ai
i 1
The sum of all the terms of the infinite sequence is called an infinite series and is denoted by

a1  a2  a3  ...   ai
i 1