1series and sequence

8/8/2019 1series and Sequence

1/16

CHAPTER 4

INFINITE SERIES

1


2/16

Objectives

` At the end of this chapter, students should be able to` Explain the definition and concepts of series, sequence and

infinite series

` Examine the convergent and divergent pf the series

` Find the radius of convergence of the series` Simplify the special Power series

` Find the Taylor and MacLaurin series

` Find the Laurent series

2


3/16


4/16

4

INFINITE SEQUENCES

` An infinite sequence is often defined by stating a formulafor the nth term, an by using {an}.

` Example:

` The sequence has nth term .

` Using the sequence notation, we write this sequence as follows

1 2 3 152 ,2 ,2 , , 2 , 2 ,nK K K

{2 }n 2n

na !

First three terms Fifth teen term


5/16

5

EXERCISE 1 :

Findingterms of a sequence

` List the first four terms and tenth term of each

sequence:

_ a3n n

2

1

13 1

n n

n

1

n

n

_ a2 0.1 n

A

B

C

D

E

F _ a4_ a2 1n


6/16

6

RECURSIVELY DEFINED SEQUENCES

` A sequence is said to be defined recursively if the firstterm a1 is state together with a rule for obtaining anyterm ak +1 from the preceding term ak whenever k 1.

` Example:

` A sequence is defined recursively as follows

` Thus the sequence is 3, 6, 12, 24, where

3

1 4 3

2 1

2 1

3 2

3 2 2 2 2 3 2 3 24

2 2 3 6

2 2 2 3 2 3 12 2 3nn

a a a

a a

a a a

! ! ! ! !

! ! !

! ! ! ! !

M

1 13, 2 for 1

k ka a a k ! ! u


7/16

7

EXERCISE 2 :

` Write down the next three terms of the sequencegiven by the following:

1 1

1128, for

14k k

a a a k

! ! u

A

B

C1 1

5, 7 2 for 1k k

a a a k ! ! u

21 12, 3 2 for 1k k ka a a a k ! ! u


8/16

8

PERIODIC SEQUENCES

` A periodic sequence is a sequence with termswhich are repeated after a certain fixed number of

term.

` Example:

1.

2.

1 11, ,1, ,

2 2K

sin2

n

n

aT

!


9/16

9

THE SUMMMATION NOTATION

OF SEQUENCES

` The symbol (sigma) is called the summation sign.

` This symbol will represents the sum of the first mterms as follows:

1 2

1

m

k m

k

a a a a

! ! KThe upper limit

The lower limit

Index of summation


10/16

10

EXERCISE 3 : Evaluating a sum

` Find the following sum:

6

1

3

1k k!

A

B

C 3

0

3 2k

k!

4

2

1

3k

k k

!


11/16

11

SERIES

` In general, given any infinite sequence,

a1, a2,an, the expression

is called an infinite series or simply a series.

1 2

1

n n

n

a a a a

g

!

! K K


12/16

12

EXERCISE 4 :

Evaluating a series` Find the nth term, the number of terms and express

each the following series by using the sigmanotation.

2 9 28 1001 KA

B

C

1 3 2 4 3 5 4 6 20 22v v v v vK

7 7 7 7

1 2 2 3 3 4 50 51 K


13/16

13

EXERCISE 5 : Express series by

sigma notation

` Write down all the terms for each of the followingseries and hence, find its sum:

6

1

2 3n

n

!A

B 8

1

2

1 2n

n

r

!

100

1

5

n!C


14/16

14

THEOREM OF SUMS

1 1 1

n n n

k k k k

k k k

a b a b

! ! !

!

1 1 1

n n n

k k k k

k k k

a b a b! ! !

!

1 1

n n

k k

k k

ca c a

! !

!

1

n

k

c nc

!

! 1n

k m

c n m c

!

! Sum of a constant

Sum of 2 infinitesequences


15/16

15

SEQUENCE OF PARTIAL SUMS

` Ifn is positive integer, then the sum of the first nterms of an infinite sequence will be denoted by Sn.

` The sequence S1, S2,Sn, is called a sequence ofpartial sums.

1 2

1

n

n n n

k

S a a a a

!

! ! Knth partial sum

1n n nS S a

!


16/16

16

EXERCISE 6: Finding the termof the sequence of partial sums

` Find the first four terms and the nth term of thesequence of partial sums associated with thefollowing sequence of positive integers.

A 1, 2, 3

, ,n,

Bg

!

1

2

1

k

k

1series and sequence

Documents