Chapter 1
Sequences
and series
Section I
Sequences
Sequences
Example
• There are two ways of defining a sequence:
(1) explicitly by giving the nth term of the
sequence; ex. a  3n  4
n
 1, 2, 5, 8, 11, 
(2) recursively by giving the first term(s) of the
sequence and a recursive formula for calculating
the remaining terms
ex.
an  an 1  6 and a1  14
14, 8, 2,  4,  10, 
Arithmetic Sequences
Example
Geometric Sequences
Example
Example
Monotonocity
Example
A Bounded Sequence
Example
Limit of a sequence
Definition of the Limit of a Sequence
Properties of Limits of Sequences
Bounded monotonic Sequences
Frequently used limits
Exercices
Find the limit of the following sequences :
Section II
Series and convergence
Series
Example
Partial sum for arithmetic and
geometric sequences
Example
Example
Convergent and divergent series
Example
Convergence of a Geometric Series
Example
Alternating series and convergence
Example
Quotient or ratio criterion
Example
Example
Root criterion
Example
Example
Others properties of Infinite Series
Direct Comparison Test
Example
Limit Comparison Test