Infinite sequence and series, Study Guides, Projects, Research of Mathematics

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INFINITE SEQUENCES AND SERIESINFINITE SEQUENCES AND SERIES

Infinite sequences and series were

introduced briefly in A Preview of Calculus

in connection with Zeno’s paradoxes and

the decimal representation of numbers.

INFINITE SEQUENCES AND SERIES

We will pursue his idea in Section 11.

in order to integrate such functions as e

• x

2



Recall that we have previously been unable

to do this.

INFINITE SEQUENCES AND SERIES

Many of the functions that arise in

mathematical physics and chemistry,

such as Bessel functions, are defined

as sums of series.



It is important to be familiar with

the basic concepts of convergence

of infinite sequences and series.

INFINITE SEQUENCES AND SERIES

11.

Sequences

In this section, we will learn about:

Various concepts related to sequences.

INFINITE SEQUENCES AND SERIES

SEQUENCE

A sequence can be thought of as a list

of numbers written in a definite order:

a

1

, a

2

, a

3

, a

4

, …, a

n

 The number a

1

is called the first term ,

a

2

is the second term , and in general

a

n

is the n th term_._

SEQUENCES

Notice that, for every positive integer n ,

there is a corresponding number a

n

So, a sequence can be defined as:

 A function whose domain is the set of positive

integers

SEQUENCES

However, we usually write a

n

instead

of the function notation f ( n ) for the value

of the function at the number n.

SEQUENCES

Some sequences can be

defined by giving a formula

for the n th term.

Example 1

SEQUENCES

In the following examples, we give three

descriptions of the sequence:

1. Using the preceding notation
2. Using the defining formula
3. Writing out the terms of the sequence

Example 1

SEQUENCES Example 1 b

Preceding

Notation

Defining

Formula

Terms of

Sequence

( 1) ( 1)

3

n

n

 (^)  n  

 

 

( 1) ( 1)

3

n

n n

n

a

 



2 3 4 5

, , , ,...,

3 9 27 81

( 1) ( 1)

,...

3

n

n

n

 

 

 

 

 

 

 

   

SEQUENCES Example 1 c

Preceding

Notation

Defining

Formula

Terms of

Sequence

 

3

3

n

n







3

( 3)

n

a n

n

 



 

0,1, 2, 3,..., n  3,...

SEQUENCES

Find a formula for the general term a

n

of the sequence

assuming the pattern of the first few terms

continues.

Example 2

SEQUENCES

We are given that:

1 2 3

4 5

3 4 5

5 25 125

6 7

625 3125

a a a

a a

  

 

Example 2