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Sequences and Series
Factorial Notation
If n is a positive number, n factorial is defined as
n !  1 2  3    n  1 n
with 0 !  1
For example,
4 !  4  3  2  1  24
Sequences and Series
An infinite sequence is a list of numbers in a particular
order.
The terms of a sequence are denoted as
a1 , a 2 , a3 , ...an , ...
Sequences and Series
Summation Notation
The sum of the first n terms of a sequence is written as
a1  a 2  a 3  ...  a n 
n
a
k 1
k
Sequences and Series
An infinite series is the sum of the numbers in an
infinite sequence.
a1  a 2  a 3  ...  a n  ... 

a
k 1
k
Sequences and Series
Arithmetic Sequences
A sequence is arithmetic if the difference between
consecutive terms is constant.
a2  a1  a3  a2  a4  a3  ...  d
d is the common difference of the series.
Sequences and Series
Geometric Sequences
A sequence is geometric if the ratio of consecutive terms is
constant.
a 2 a 3 a4


 ...  r
a1 a2 a3
r0
r is the common ratio of the series.
Sequences and Series
Geometric Series
The sum of the terms in an infinite geometric sequence is
called a geometric series.

a  ar  ar  ar  ...  ar  ...   ar k
2
3
n
k 0
If r  1 , the series has the sum
a
S
1 r