Week 1infinite Series

8/12/2019 Week 1infinite Series

1/30

1

CHAPTER 1INFINITE SERIES


2/30

2

INFINITE SERIESWeek 1 (a)

Todays Objective :

Determine and contrast the Taylor and Maclaurin

polynomials of functions


3/30

3

INFINITE SERIES

An infinite seriesis an expression of the form

and the nth partial sum of the series is

If the sequence of partial sums converges to a limit L,

1 2 3

1

... kk

a a a a

1 2 3

1

...n

n n k

k

S a a a a a

1 2 3

1

...n

n n k

k

S a a a a a L

Example: Bouncing Ball

You drop a ball from a meter above a flat surface. Each time the ball hits

the surface after falling a distance h, it rebounds a distance rh, where r is

positive but less than 1. Find the total distance the ball travels up and down


4/30

4

INFINITE SERIES

Taylor and Maclaurin Series

Taylor series atx=cfollows the Maclaurin series for each function.

2 3

... ...2! 3! !

kc c c

x c c e x c e x c e x c

e e e x ck

2 3

cos cos sin cos sin ...2! 3!

x c x c

x c x c c c c

2 3

sin sin cos sin cos ...2! 3!

x c x cx c x c c c c

2 3 1

2 3

11n 1n ... ...

2 3

k k

k

x c x c x cx cx c

c c c kc

2 3 41 1 ... ...1

ku u u u uu

And lots more

Exponential Series

Cosine Series

Sine Series

Logarithmic Series

Geometric Series


5/30

5

Taylor and Maclaurin

Polynomials


6/30

6

Maclaurin Polynomials for f(x) = s in(x).

(Also known as Taylor polynomials atx=0)


7/30

0P x 0P x x

3

03!

xP x x

3 5

03! 5!

x xP x x

3 5 7

03! 5! 7!

x x xP x x

3 5 7 9

0 3! 5! 7! 9!

x x x xP x x

3 5 7 9 11

03! 5! 7! 9! 11!

x x x x xP x x

3 5 7 9 11 13

0 3! 5! 7! 9! 11! 13!

x x x x x xP x x

sinf x x


8/308

Maclaurin Polynomials for f(x) = cos(x).



9/30

1P x 2

12!

xP x

2 4

12! 4!

x xP x

2 4 6

12! 4! 6!

x x xP x

2 4 6 8

12! 4! 6! 8!

x x x xP x

2 4 6 8 10

1 2! 4! 6! 8! 10!

x x x x xP x

2 4 6 8 10 12

1 2! 4! 6! 8! 10! 12!

x x x x x x

P x

2 4 6 8 10 12 14

1 2! 4! 6! 8! 10! 12! 14!

x x x x x x x

P x

cosf x x


10/3010

Maclaurin Polynomials for f(x) = e(x).



11/30

1P x 11!

xP x

11! 2!

x xP x 1

1! 2! 3!

x x xP x 1

1! 2! 3! 4!

x x x xP x

1 1! 2! 3! 4! 5!x x x x x

P x 1 1! 2! 3! 4! 5! 6!x x x x x x

P x 1 1! 2! 3! 4! 5! 6! 7!x x x x x x x

P x

xf x e


12/3012

Taylor and Maclaurin Polynomials.

http://mathdemos.gcsu.edu/mathdemos/TaylorPolynomials/index.html
http://mathdemos.gcsu.edu/mathdemos/TaylorPolynomials/index.htmlhttp://mathdemos.gcsu.edu/mathdemos/TaylorPolynomials/index.html


13/30

13

INFINITE SERIESWeek 1 (b)

Objective :

Derive the Taylor and Maclaurin series for functions, and its

radius of convergence.


14/30

14


Suppose there is an open intervalIcontaining cthroughout which

the functionfand all its derivatives exist. Then the power series

is called the Taylor series of fat c.

The special case where c = 0is called the

Maclaurin series of f:

2 3

...1! 2! 3!

f c f c f cf c x c x c x c

2 30 0 00 ...

1! 2! 3!

f f ff x x x


15/30

15

Taylor series atx=c. (Remember that for Maclaurin seriesx=0)

2 3

... ...2! 3! !

kc c c

x c c e x c e x c e x c

e e e x ck

2 3

cos cos sin cos sin ...2! 3!

x c x c

x c x c c c c

2 3

sin sin cos sin cos ...2! 3!

x c x cx c x c c c c

2 3 1

2 3

11n 1n ... ...

2 3

k k

k

x c x c x cx cx c

c c c kc

2 3 41 1 ... ...1

ku u u u uu

And lots more


Exponential Series

Cosine Series

Sine Series

Logarithmic Series

Geometric Series


16/30

16


Find the Maclaurin series for cosf x x

Solution :1. Set c=0(since it is Maclaurin series)

2. Note that fis infinitely differentiable at x=0

4 4

2 3

22 4 6

0

cos 0 1

sin 0 0cos 0 1

sin 0 0

cos 0 1

0 0 0cos 0 ...

1! 2! 3!

1cos 1 ...

2! 4! 6! 2 !

k k

k

f x x f

f x x ff x x f

f x x f

f x x f

f f fx f x x x

xx x xx

k


17/30


18/30

18


How to define its

Radius of Convergence ?


19/30

19


The Ratio Test

1

1

1 1

1

(i) If lim 1, then the series is abso1ute1y convergent

(and therefore convergent).

(ii) If lim 1 or lim , then the series is divergent.

(iii) If lim

nn

nnn

n nn

n nnn n

n

aL a

a

a aL a

a a

1 1, the Ratio Test is inconc1usive; that is, no conc1usion

can be drawn about the convergence or divergence of .

n

n

n

a

a

a

We used ratio test because involved k!, kp, or ck


20/30

20


Radius of Convergence

(i) The series converges on1y when .

The radius of convergence is 0 .

(ii) The series converges for all .

The radius of convergence is .

(iii) There is a positive number such that the

x a

R

x

R

R

series convergesif and diverges if .x a R x a R


21/30

21



0

1

1

Examp1e 1: ! , 1et ! . Then

1 !

1 as 0 .!

By the Ratio Test, the series diverges when 0.

Thus, the given s

n n

n

n

n

n

nn

n x a n x

n xa

n x n xa n x

x

eries converges on1y when 0.

Then, radius of convergence, 0.

x

R


22/30

22



0

1

1

1

3 3Examp1e 2: , 1et . Then

3 3

1 1 13

1 3 3 as

1

By the Ratio Test, the series is conve

n n

n

n

n n

n

n

n

n

x xa

n n

x xa n n

a n nx

x x n

rgent,

when 3 1 and divergent when 3 1

Then, radius of convergence, 1.

x x

R


23/30

23



2 2

2 22 20

1 22 1 2 2

1

2 22 22 1

2

2

1 1Examp1e 3: , 1et . Then

2 ! 2 !

1 2 ! 11 2 12 1 !

0 as

4 1

By th

n nn n

nn nn

n n n

n

n nnn

x xa

n n

x n xaa x nn

xn

n

e Ratio Test, the series converges for a11 .

Then, radius of convergence, .

x

R


24/30

24


Find the Radius of Convergence for :(Note that, you must determine the Taylor series for below functions first)

1. at 2

2. at 0 (Mac1aurin Series)

3. sin at 0 (Mac1aurin Series)

4. sin centered at3

x

x

f x e c

f x e c

f x x c

f x x


25/30


26/30

26


Questions :

Proved that :

1. The Maclaurin series for

2. The Taylor series at x = cfor

2 31 1 ... ...1

kx x x xx

2 3 1

2 3 11n 1n ... ...2 3

k k

kx c x c x cx cx c c c c kc


27/30

27


Now, we look on the

Operations with



28/30


29/30

29


2

5 21.

3 2

12. 1n

1

3. cos

xf x

x

x

x

x

Hint: Use long division and

modifying a geometric series


30/30

30

THE END

DO YOUR TUTORIAL !!

Week 1infinite Series

Documents

Transcript of Week 1infinite Series