Sequences

Lesson 8.1

Definition

• A __________________ of numbers• Listed according to a given ___________________• Typically written as a1, a2, … an

• Often shortened to { an }

• Example• 1, 3, 5, 7, 9, … • A sequence of ______________ numbers

Finding the nth Term

• We often give an expression of the general term

• That is used to find a specific term• What is the 5th term of the above sequence?

Sequence As A Function

• Define { an } as a ____________________• Domain set of nonnegative _______________• Range subset of the real numbers• Values a1, a2, … called _________of the sequence

• Nth term an called the general term

• Some sequences have limits• Consider ( ) lim ( ) ?

1nn

nf n a f n

n

Converging Sequences

• Note Theorem 9.2 on limits of sequences• Limit of the sum = sum of limits, etc.

• Finding limit of convergent sequence• Use table of values• Use ________________• Use knowledge of rational functions• Use ___________________________Rule

2 1

3 4

n

n

5/2 n

Divergent Sequences

• Some sequences ___________

• Others just grow __________________

sin2

nn

2 5 3

2

n n

n

Determining Convergence

• Manipulate algebraically

• ___________________and take the limit

2

2 2

2

33 3

3

n n nn n n n n n

n n n

conjugate expressions

conjugate expressions

Determining Convergence• Consider

• Use l'Hôpital's rule to _______________________________of the function

• Note we are relating limit of a sequence from the limit of a ________________ function

2

1 n

n

e

2

0

2 2lim lim lim 0

1 x x xx n n

x x

e e e

Bounded, Monotonic Sequences

• Note difference between• Increasing (decreasing) sequence• __________________ increasing (decreasing)

sequence• Table pg 500

• Note concept of bounded sequence• Above• Below Bounded implies ________________• Both

Assignment

• Lesson 9.1• Page 602• Exercises 1 – 93 EOO

Series and Convergence

Lesson 9.2

Definition of Series

• Consider summing the terms of an infinite sequence

• We often look at a _______________sum of n terms

1 2 ... ...nS a a a

1 2 ... ___________n nS a a a

Definition of Series

• We can also look at a _____________of partial sums { Sn }

• The series can _________________________• The sequence of partial sums converges• If the sequence { Sn } does not converge, the series

diverges and has no sum

1

limk nnk

a S S

Examples

• Convergent

• Divergent

0

2 4 14 40 1222, , , , , ...

3 9 27 813k

k

1

1 1 1 1 1 1 1 ...

1 if n is odd

0 if n is even

k

k

nS

Telescoping Series

• Consider the series

• Note how these could be regrouped and the end result

• As n gets large, the series = 1

1

1 1 1 1 1 1 11 ...

1 2 2 3 3 4n n n

Geometric Series

• Definition• An infinite series• The ______________of successive terms in the

series is a ________________

• Example• What is r ?

2 3

0

... ...k n

k

a r a a r a r a r a r

0

2 2 2 22 ...

3 9 273k

k

Properties of Infinite Series

• ______________________• The series of a sum = the sum of the series

• Use the property

k k k kc a d b c a d b

2 20 0 0

2 5 1 12 5

3 3k k

k k kk k

Geometric Series Theorem

• Given geometric series(with a ≠ 0)

• Series will• Diverge when | r | _________• _______________when | r | < 1

• Examples• Compound

interest

0

k

k

a r

1

100 1.05k

k

Or

Applications

• A pendulum is released throughan arc of length 20 cmfrom vertical

• Allowed to swing freelyuntil stop, each swing 90%as far as preceding swing past vertical

• How far will it traveluntil it comes to rest?

20 cm

Assignment

• Lesson 9.2• Page 612• Exercises 1 – 69 EOO

The Integral Test; p-Series

Lesson 9.3

Divergence Test

• Be careful not to confuse• Sequence of general terms { ak }

• Sequence of partial sums { Sk }

• We need the distinction for the divergence test• If

• Then must _________

lim 0kka

ka

Note this only tells us about ______________. It says nothing about convergence

Note this only tells us about ______________. It says nothing about convergence

Convergence Criterion

• Given a series

• If { Sk } is _________________________• Then the series converges• Otherwise it diverges

• Note• Often difficult to apply• Not easy to determine { Sk } is bounded above

with 0 for all k ka a k

The Integral Test

• Given ak = f(k)• k = 1, 2, …• f is positive, continuous, _____________for x ≥ 1

• Then either

• both converge … or• both _________________

1 1

( )kk

a and f x dx

Try It Out

• Given

• Does it converge or diverge?

• Consider

3/ 2

1

2k

k

3/ 2

1

lim 2b

bx dx

p-Series

• Definition• A series of the form• Where

p is a _____________________

p-Series test• Converges if _____________• ___________if 0 ≤ p ≤ 1

1

1 1 1 1...

1 2 3p p p pk k

1

1p

k k

Try It Out

• Given series

• Use the p-series test to determine if it converges or diverges

1

k

k

e

Assignment

• Lesson 9.3• Page 620• Exercises 1 - 35odd

Comparison Tests

Lesson 8.4

Direct Comparison Test

• Given

• If converges, then converges

• What if

• What could you conclude about these?

0 for all for some k ka c k N N

1k

k

c

0 for all k Nk kd a

1

andkk

d

1k

k

a

Try It on These

• Test for convergence, divergence• Make comparisons with a geometric series or

p-series

1

0.5k

k

/ 2

0

2k

k

Limit Comparison Test

• Given ak > 0 and bk > 0 for all sufficiently large k … and …

where L is finite and positive• Then

either both ___________… or both _________

1

k

k k

aL

b

k and bka

Limit Comparison Test

Strategy for evaluating

1. Find series with _______________ and general term "essentially same"

2. Verify that this limit exists and is positive

3. Now you know that _________________ as

kakb

lim k

nk

a

b

kakb

Example of Limit Comparison

• Convergent or divergent?

• Find a p-series which is similar

• Consider

• Now apply the comparison

1

1

1k k k

Assignment

• Lesson 9.4• Page 628• Exercises 1 - 27 odd

Taylor and MacLaurin Series

Lesson 9.7

Taylor & Maclaurin Polynomials

• Consider a function f(x) that can be differentiated n times on some interval I

• Our goal: find a _____________function M(x)• which approximates f • at a number c in its domain

• Initial requirements• M(c) = ____________• ____________ = f '(c)

Centered at c or ____________Centered at c or ____________

Linear Approximations

• The ____________________is a good approximation of f(x) for x near a

a x

f(a)

f'(a) (x – a)

(x – a)

Approx. value of f(x)Approx. value of f(x)True value f(x)True value f(x)

1( ) ( ) ( ) '( ) ( )f x M x f a f a x a

Linear Approximations

• Taylor polynomial degree 1• Approximating f(x) for x near 0

• Consider• How close are

these?• f(.05)• f(0.4)

1(0) ( ) (0) '(0)f M x f f x

( ) cos 1 for x near 0f x x

Quadratic Approximations

• For a more accurate approximation to f(x) = cos x for x near 0• Use a __________________ function

• We determine

• At x = 0 we must have• The functions to agree • The first and ________________________ to agree

22 0 1 2( )M x a a x a x

Quadratic Approximations

• Since

• We have

22 0 1 2

2 1 2

2 2

( ) and ( ) cos

' ( ) 2 and '( ) ________

" ( ) 2 and ''( ) ________

M x a a x a x f x x

M x a a x f x

M x a f x

0 2 0

1 2 1

2 2 2

(0) (0) cos0 1 so 1

' (0) '(0) sin 0 0 so 0

2 " (0) "(0) cos0 1 so _____

a M f a

a M f a

a M f a

Quadratic Approximations

• So

• Now how close are these?• •

22

1cos ( ) 1 0

2x M x x x

2

2

(.05) (.05)

(0.4) (0.4)

f M

f M

Taylor Polynomial Degree 2

• In general we find the approximation off(x) for x near 0

• Try for a different function• f(x) = sin(x)• Let x = 0.3

2( ) ( ) ____________________f x M x

Higher Degree Taylor Polynomial

• For approximating f(x) for x near 0

• Note for f(x) = sin x, Taylor Polynomial of degree 7

( )2

( ) ( )

"(0) '''(0) (0)(0) '(0) ...

2! 3! !

n

nn

f x M x

f f ff f x x x

n

2 3

7

4 5 6 7

sin ( ) 0 02! 3!

0 04! 5! 6! 7!

x xx M x x

x x x x

Improved Approximating

• We can choose some other value for x, say x = c

• Then for f(x) = ex the nth degree Taylor polynomial at __________

2 3

( )1! 2! 3! !

n

c c c c cn

x c x c x cx cT x e e e e e

n

2 3

( )1! 2! 3! !

n

c c c c cn

x c x c x cx cT x e e e e e

n

Assignment

• Lesson 9.7A• Page 656• Exercises 1 – 23 odd

• Lesson 9.7B• Page 656• Exercises 25 – 43 odd