1 Sequences and Series From Simple Patterns to Elegant and Profound Mathematics David W. Stephens The Bryn Mawr School Baltimore, Maryland PCTM – 28 October.

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Sequences and SeriesFrom Simple Patterns to Elegant and Profound Mathematics

David W. Stephens

The Bryn Mawr School

Baltimore, Maryland

PCTM – 28 October 2005

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Contact Information

Email:[email protected]

The post office mailing address is:David W. Stephens109 W. Melrose AvenueBaltimore, MD 21210410-323-8800

The PowerPoint slides will be available on my school website:

http://207.239.98.140/UpperSchool/math/stephensd/StephensFirstPage.htm , listed under “PCTM October 2005”

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Teaching Sequences and Series

We will look at some ideas for teaching sequences and series as well as some applications in mathematics classes at THREE different levels:

I. Early (Algebra 1, Algebra 2, and Geometry)

II. Intermediate (Advanced Algebra, Precalculus)

III. Advanced (AP Calculus, esp. BC Calculus)

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Teaching Sequences and Series

Many of the topics and examples used today will not be new to you, but I want you to consider thinking of them …and talking about them with students … as sequences and series. It can be a good way for them to think about these diverse topics as bring linked mathematically. Just as functions link a lot of what we teach, the patterns of sequences and series can tie these ideas together for better comprehension.

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Early Sequences and Series (Algebra 1, Algebra 2, and Geometry)

1. Looking for patterns

2. Identifying kinds of sequences

3. Describing patterns in sequences

4. Using variables

5. Summation notation

6. Strategies for summing

7. Applications with geometry ideas

8. Graphing patterns

9. Data analysis & functions as sequences

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Intermediate Sequences and Series

(Advanced Algebra, Precalculus)

1. More geometric sequences and exponential functions

2. Infinite series

3. Convergence and divergence

4. Informal limits

5. More advanced data analysis (“straightening data”)

6. Applications (compound interest, astronomy, chemistry, biology, economics, periodic motion or repeating phenomena)

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Advanced Sequences and Series (AP Calculus, esp. BC Calculus)

1. Newton’s method for locating roots

2. Riemann sums

3. Trapezoid rule and Simpson’s rule

4. Euler’s method for differential equations

5. Power series (Maclaurin and Taylor series polynomials)

6. Convergence tests for series

Early Sequences and Series 8

Early Topics (Algebra 1 , Algebra 2, and Geometry)

I sometimes have begun my Algebra 2 classes in September with this topic because…

a) New students to the school (and the class) do not feel

“new”.

b) I can use algebraic language.

c) I can review linear functions in a new context.

d) I can sneak in some review which does not feel like

review!


Activity 1

Find the next three numbers in these sequences:

A) 6, 9, 13, 18, 24, …B) 12, 17, 13, 16, 14, 15, 15, …C) 5, 10, 20, 40, …D) 7, -21, 63, -189, …E) 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, 0, 1, 2, 3, …


Activity 2

Students build their own sequences, and they challenge their classmates to guess the next few entries. This can be a neat homework assignment. (It can be extended to later activities where they have to code their sequence patterns with variables, too.)


Activity 3

Describe the pattern in words:

A) 7, 5, 3, 1, -1, -3, …

B) 70, 68, 64, 56, 40, 32, 28, 26, 25, …

C)

D) 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 14, 15, ...

E) 1, 2, 3, 4, 4,3, 2, 1, 3, 5, 7, 7 , 5, 3, 1, 12, 23, 34, 23, …

F) 4, 9, 32, 50, 53, 54, 54, 54, 54, 54.1, 54.13, 54.135,

54.1356 , ...

,...17

81,

13

7,

11

5,

9

4,

8

3,

7

2,

4

1


Activity 4

Learn to code the pattern with variables:

A) 9, 13, 17, 21, 25, 29 , …

Let a0 = 9 an = an-1+ 4 or an = a0+ (n-1)d

(Some texts use tn, where t = term, instead of an)

It could also be coded that a6 = 9 and then a7 = 13,

if you decided to start the count at item #6


Activity 4 (continued)

B) 3, 6, 12, 24, 48, …

Let a0 = 3 an = a0r n-1

= (first term)(ratioterm# - 1)

( or let t0 = 3 tn = a0r n-1)


Activity 5

Introduction to Fibonacci sequences

A) 1, 1, 2, 3, 5, 8, 13, 21, 34

B) 2, 5, 7, 12, 19, 31, 50

an = an-1+ an-2 (recursively defined functions)


Activity 6

Series and Summation Notation

1. a) sequence 1 , 3 , 5 , 7 , 9 , 11

b) series 1 + 3 + 5 + 7 + 9 + 11

c) series notation or or 2n 1n1

6

2n 1n0

5

2n 9n5

10


Activity 6 (continued)

2. a) sequence 2, 6, 18, 54

b) series 2 + 6 + 18 + 54

c) series notation

3. a) sequence 4, -2, 1, , …

b) series 4 – 2 + 1 - + - …

c) series notation

2n

n1

4


Activity 7

Interleaved and other creative sequences

Find the next three terms, and describe the two sequences that are interleaved.

A) 1, 3, 4, 9, 7, 27, 10, 81, 13, 243 , …1, 3, 4, 9, 7, 27, 10, 81, 13, 243 , …

B) 5, 1, 7, 4, 9, 7, 11, 10, 13, 13, 5, 1, 7, 4, 9, 7, 11, 10, 13, 13, …


Introduction to Series

How to add up an arithmetic series efficiently:

Example: sn = 6 + 9 + 12 + 15 + 18 + … + 219

• Add the first and last terms, the second and the second to the last, etc. What do you notice?

• How many pairs are there? • What if there are an odd number of terms to

add? Sn =

n

2(s

1 s

n)


Introduction to Series

How to add up a geometric series efficiently:

Example: sn = 5 + 10 + 20 + 40 + 80 + 160 + 320= a0 + a0r + a0r2 + a0r3+ … + a0rn-1

rsn = a0r + a0r2 + a0r3 + a0r4 + … +a0rn

Then sn – rsn = a0 – a0rn

This provides us with the usual formula for a geometric series:

sn =

0 (1 )

1

na r

r


Activity 8

For the series,

sn = 5 + 10 + 20 + 40 + … + 320 + …

calculate s28


Write an arithmetic series and a geometric series so that the value of the sum for the arithmetic series is greater than the sum for the geometric series for the first 10 terms, but a) the arithmetic series still exceeds the

geometric series for the first 20 terms b) the geometric series exceeds the arithmetic

series for the first 20 terms

Activity 9


Find specific terms in sequences a) For 6, 11, 16, 21, … calculate t41 b) For an arithmetic sequence with t1 = 7 and t4 = 16, calculate t 91 c) For an arithmetic sequence with t4 = -9 and t6 = 7, calculate t50 d) For a geometric sequence with t1 = 9 and r = .49, calculate t102

Activity 10


Treat the list of x and y coordinates as sequences! Example 1: x 1 2 3 4 5 6 7 8 y 7 10 13 16 19 22 25 28 x is an arithmetic sequence y is an arithmetic sequence This is sometimes called an “add-add” property. Thus, y = f(x) is LINEAR What is the actual function? Ans: f(x) = 3x + 4 3 = the common difference = d in “sequence language” 4 = a0 in the sequence

Data Analysis

Building Functions from Data


Example 2: x 3 5 7 9 11 13 15 17 y 4 8 12 16 20 24 28 32 x is an arithmetic sequence y is an arithmetic sequence Thus f(x) is LINEAR again. What is the actual function? Ans: common difference = d = 2 = slope If a3 = 4 and d = 2, then a0 = -2 So f(x) = 2x - 2

Data Analysis



Example 3: x 1 2 3 4 5 6 7 8 y 6 9 14 21 30 41 54 69 x is an arithmetic sequence y is an arithmetic sequence, and y is constant Thus f(x) is QUADRATIC. What is the actual function? Ans: f(x) = x2 +5

Data Analysis



GeometryAngles of Polygons

What is the general formula for the sum of the interior angles of a polygon with n sides?

(n, measures of interior angles) :

(3, 180) , (4, 360), (5 , 540) , (6 , 720) , …

(n , 180(n-2))


GeometryA Modeling Application

Handshake Problem:

If n people shakes hands with everyone else at a meeting, how many handshakes occur?1. Visualize this as a geometry problem.2. Consider a simpler version with just a few

number of people.3. Generalize the data, and consider the data

as sequence.



Handshake Problem:

A A B

C

DE

F

29


n = number of people

h(n) = number of handshakes

n 1 2 3 4 5 67

h(n) 0 1 3 6 10 1521


GeometryA Derivation of

Find the perimeter of a sequence of regular polygons which are inscribed in a unit circle, and emphasize that the sequence of results is important to watch.

s = length of one side of the polygon

p = perimeter of the polygon



2

-2

A

C D

B

E

2

-2

A

C DBE

s =

p = 3 = 5.196

3

3

s =

p = 4 = 5.657

2

2

s = length of one side

p = perimeter of inscribed polygon



2

-2

A

C DE

2

-2

H

AE F

IG

s = 2 sin(36) = 1.176

p = 5(1.176) = 5.878

s = 1

p = 6



)360

*5sin(.n

)180

sin(2n

In general, the length of one-half of a side of an inscribed regular polygon is

So a side measures and the

perimeter of the polygon measures

Since p 2 , then can be calculated.

)180

sin(2n

n



1

0.5

-0.5

-1

-1.5

-1 1

A

IG

n

360The central angle

for each side is

Each half-side has length equal to the sine of one-half the central angle.



Here are the perimeters of the polygons from the TI-83 as a list (L2)

Note: Ignore L3.

Intermediate Sequences and Series 36

Intermediate Sequences and Series (Advanced Algebra, Precalculus)

1. More geometric sequences and exponential functions

2. Infinite series

3. Convergence and divergence

4. Informal limits

5. More advanced data analysis (“straightening data”)

6. Applications (compound interest, astronomy, chemistry, biology, economics, periodic motion or repeating phenomena)


Data AnalysisBuilding Functions from Data

Example 4: x 1 2 3 4 5 6 7 8

y 3 9 27 81 243 729 2187 6561

x is an arithmetic sequence

y is a geometric sequence

This is sometimes called an “add-multiply” property

So y = f(x) is EXPONENTIAL

What is the actual function?

Ans: f(x) = 3x

( where r = 3 in the geometric sequence)



Example 5: x 1 2 3 4 5 6 7 8 y 5 11 29 83 245 731 2189 6563

y 3 9 27 81 243 729 2187 6561

x is an arithmetic sequence y is not exactly a geometric sequence

But if the sequence of y-values is compared with the last set of y’s, then we see that this sequence is 2 more than a geometric sequence.

So y = 3x + 2



Example 6: x 1 4 7 10 13 y 6 48 384 3072 24,576

x is an arithmetic sequence y is not exactly a geometric sequence

Since the two sequences have the “add-multiply” property, then y is a geometric sequence, and it is exponential. Notice that the x’s do not have to be consecutive.

We have to find the “r” value as if we are calculating geometric means



Example 7: x 1 4 7 10 13

y 6 48 384 307224,576

a1 = 6 and a4 = 48, and we need to fill in the sequence so that we know the y-values for terms 2 and 3. Since the desired sequence is geometric, we need to know what to multiply a1 by repeatedly three times to get 48. This suggests that r*r*r= 48/6.

So r = = 2 , and y = 3 * 2x

3 8



Example 7:

x 1 2 3 4 5 6 7 8 9 10 11

y 6 12 24 48 96 192 384 768 1536 3072 6144

r = = 2 , and y = 3 * 2x 3 8


An Historical Diversion

Let’s take a look at the pairing of an arithmetic and a geometric sequence.

n 1 2 3 4 5 6

an 2 4 8 16 32 64

Let’s suppose that we wanted intermediate terms:

n 1 3/2 2 5/2 3 7/2 4 9/2 5 6

an 2 4 8 16 32 64



n 1 3/2 2 5/2 3 7/2 4 9/2 5

an 2 4 8 16 32

Thinking about an as a geometric sequence, we need a geometric mean to fill in the missing terms. Our desired multiplier, r, is .

an 2 4 8 16 32

an 2 2 3/2 4 2 5/2 8 2 7/2 16 2 9/2 32

2

22 24 28 216



So when we write an = 2n , then the sequence, n, becomes the exponents, or the logarithms, for the geometric sequence.

This is part of the history of Henry Briggs, John Napier, Jobst Burgi, John Wallis, and Johann Bernoulli from 1620 to 1749 in the development of logarithms.

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X 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

g(x) 11 13 5 6 3 8 9 14 10 21

Function Transformations using SequencesIf functions are considered as lists of data, and one function is a

transformation of another one, then the alterations to the sequence of function values is the key to decoding the transformation.

X 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

-3 5 1 5 -4

g(x) 11 13 5 6 3 8 9 14 10 21

-3 5 1 5 -4

We want to write g(x) as a transformation of f(x), so g(x) = f(x – 3)

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Function Transformations using Sequences

Preliminary questions:

A. When a transformation such as f(x + a) is used, what happens to the y values?

B. When a transformation such as f(x) + a is used, what happens to the y values?

C. When a transformation such as a*f(x) is used, what happens to the y values?

X 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

-3 5 1 5

g(x) 4 8 -1 0 6 7 4 9 10 15

-3 5 1 5

X 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

g(x) 4 8 -1 0 6 7 4 9 10 15

g(x) = f(x – 5) + 1

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Function Transformations using Sequences

x 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

g(x) -8 3 12 6 16 18 28 20 42 86

g(x) = 2f(x – 2)

x 0 1 2 3 4 5 6 7 8 9

f(x) 6 3 8 9 14 10 21 43 8 6

-3 5 1 5 -4

g(x) -8 3 12 6 16 18 28 20 42 86

-6 1- 2 1- -8


Infinite Sequences, Series and Convergence

There are some really good opportunities to lead students to important conclusions, as well as to challenge their intuition with some sophisticated ideas … with infinite sequences and series.

We can extend their numerical sense as well as exploiting their graphical skills to help generate conclusions.



Suppose an: 1, 3, 5, 7, 9, …

Where does an go as n gets large?

Suppose bn: 1, 1.01, 1.02 , 1.03 , 1.04 , …

Where does bn go as n gets large?

Suppose cn: 1, 2, 4, 8, 16, …

Where does cn go as n gets large?



Suppose dn:

Where does dn go as n gets large?

Since this is the ratio of two sequences, each of which approaches infinity, explain your answer to this question.

,...32

9,

16

7,

8

5,

4

3,

2

1



Suppose en: 1, 0, -1 , 1 , 0 , -1 , 1 , 0 , -1 , …Where does en go as n gets large?

Suppose fn: 40 , 32, 25.6, 20.48, 16.384, …Where does fn go as n gets large?

Type 40 on the calculator and hit ENTER. Type * .8 and hit ENTER

The screen will read ANS * .8Repeatedly hit enter to generate the sequence.


Infinite Sequences, Series and ConvergenceSuppose gn: 60, 90, 108, 120, , 135, …

Where does gn go as n gets large?What sequence is this?

Suppose hn: 120, 90, 72, 60, , 45, …

Where does hn go as n gets large?What sequence is this?

7

4128

7

351



Suppose in:

Where does in go as n gets large?

,...8343

8341,...,

100

98,...,

5

3,

4

2,

3

1,

2

0


Intermediate Level ApplicationsSequence Mode on the Calculator

Suppose we want to generate the sequence as an iterated function (recursive function).

So: 2, 5, 8, 11, 14, 17, …

could be an = 2 + 3n or an = an-1 + 3


Intermediate Level ApplicationsSequences and Series on the Calculator

To generate sequences on the HOME screen, go to LIST (2nd STAT)/OPS/<Option 5>

which will give seq(

The inputs required for seq( are:

seq(expression, variable, begin, end [increment])

Example: an = 2n+1 1, 3, 5, 7, 9, …


Intermediate Level ApplicationsSequence and Series on the Calculator

Example: an = 2n+1 1, 3, 5, 7, 9, …

Notice that the name of the variable does not matter, as long as it is specified.



If the series is desired, the sum( function is used.

Example: an = 2n+1 1, 3, 5, 7, 9

sn = sum(an) 1+3+5+7+9 = 25

“Sum(“ is found in LIST (2nd STAT)/MATH/<Option 5>



Partial sums can also be generated, and this is helpful if there is an application where the sums should be considered as making a sequence, perhaps if their convergence is being considered.

The function cumSum( is found under

LIST (2nd STAT)/OPS/<Option 6>

Example: an = 2n+1 1, 3, 5, 7, 9cumSum(an) 1, 4, 9, 16, 25



On the calculator, cumSum( {1, 3, 5, 7, 9})

If a list is already in the calculator, perhaps in L1, then cumSum(L1) or sum(L1) will give series results.

or …

cumSum(seq(2N+1, N, 0, 4))



First term value Recursive

function

First term #



Using the same recursive function:

an = an-1 + 3 or u(n) = u(n-1) +3,

suppose that we want to build a sequence in a list on the calculator.



It is also possible to use the sequence mode to graph some more complicated ideas.

Suppose that we are trying to convince a student that the geometric sequence 100, 80, 64, 51.2, … converges.

Set the Window to



Go to the 2nd ZOOM [Format] key, and make sure that TIME is selected at the top.

Hit GRAPH.

This is a scatterplot of

the (n, an)

u(n)=0.8u(n-1)



Instead, select 2nd Zoom [Format] and choose Web.

Set the Window to 0 < x< 105 and 0 < y < 105

Hit Graph and

Trace. Hit the

right arrow to

iterate the web.



Let’s look at an = 10(-.8)n,

which becomes u(n) = -.8u(n-1)

You have to think about the WINDOW, but it has a web which looks like:



An application which is stretching toward the advanced is the idea of a predator-prey model *. The populations of the two populations depend on the size of the other population. Depending on various parameters, the populations will either die out, grow without bound (!), or move into an equilibrium.

Two sequence functions can be used:

Rn = Rn-1(1+0.05 -.001*Wn-1) “rabbits”

Wn = Wn-1(1+0.0002Rn-1 – 0.03) “wolves”

(This example is from the TI-83 manual, page 6-13)

* There is a long document on my website about predator-prey models that I co-

wrote as a NSA sponsored project in June 2004.





Using a WINDOW of nMin = 0 and nMax = 400

PlotStart = 1 PlotStep =1

XMin = 0 XMax = 400 Xscl = 100

YMin = 0 YMax = 300 YScl = 100

Under FORMAT, use the TIME

choice.

It makes for great classroom discussion to interpret these graphs.



With the sequence mode, we can do something quite interesting on the calculator. The first graph was showing the separate rabbit and wolf populations as time progressed. But what if we want to see how the graphs of the two populations look relative to each other, i.e., (rabbits, wolves).

To do this select the FORMAT key and then find the uv choice at the top.



Experimentation with the data suggests that the new WINDOW be:

XMin = 80 XMax = 250 Xscl = 50

YMin = 0 YMax = 100 YScl = 10



When series go on forever, we call them infinite series. Let’s look at arithmetic series first.

Sn = 4 + 7+ 10 + 13 + …

=

What is the sum s100? s1000 ? s 1,000,000 ?

2

)( 1 naan



Compare the results for each of these arithmetic series:

1. Sn = 4 + 7 + 10 + 13 + …

2. Sn = 1 + 1.1 + 1.2 + 1.3 + 1.4 + …

3. Sn = 5+ 5.001 + 5.000001 + …

4. Sn = 4 + 3.5 + 3 + 2.5 + 2 + …

Conclusion…… ?



Moving onto geometric series, consider the behavior of these sums by taking the number of terms to be higher and higher.

1. sn = 2 + 4 + 8 + 16 + …

sn =

s10 = s100 = s1000 =

r

ra n

1

)1(1



2. sn = 2 + 2(1.02) + 2(1.02)2 + 2(1.02)3 + …

3. sn=2 + 2(0.98) + 2(.98)2+ 2(.98)3 + …

4. sn = 1 – 3 + 9 – 27 + 81 + …

5. sn = 1 +

6. sn = 1 +

...8

1

4

1

2

1

...8

1

4

1

2

1



It eventually becomes obvious that there are geometric series which converge and others which diverge.

The idea is that convergence depends on the value of r (the common ratio).

Conclusion: An infinite geometric series converges when 1 < r < 1 or |r| < 1



Looking at the sequences graphically makes some strong connections with algebra, and the visual impact helps with understanding about convergence and divergence.

Let’s look at some ideas about series first (because the graphs of sequences vs. the graphs of series is also an important distinction).


Graphs of Sequences and Series

12

2

n

Examples:

an = bn =

cn = dn =

12

3

n

2(3 )n 21

3n

8 8 8 8

1 1 1 1n n n n

n n n n

a b c d






Intermediate Level ApplicationsDeer Populations

In this application, the various quantities affect each other. This is part of a discrete mathematics topic. The sequences involved (and note why they are not series!) affect each other. Whether or not they converge is the important point, since this involves whether the populations remain stable, or whether they explode or become extinct. There are intuitive ideas of limits here.



Newborn Yearling Adult Male Adult Female TOTAL

N Y AM AF

N = 0.20 AF Y = 0.90 N AM = 0.90 AM + 0.45 Y AF = 0.90 AF + 0.48 Y

1 20 16 90 65 191

2 13 18 88 66 185

3 13 11 87 68 179

4 13 11 83 66 173

5 13 11 79 64 167

6 12 11 76 62 161

7 12 10 73 61 156

8 12 10 70 59 151

9 11 10 67 57 145

10 11 9 64 56 140



Newborn Yearling Adult Male Adult Female TOTAL

N Y AM AF

N = 0.20 AF Y = 0.90 N AM = 0.90 AM + 0.45 Y AF = 0.90 AF + 0.48 Y

11 11 9 61 54 135

12 10 9 58 52 129

13 10 9 56 51 126

14 10 9 54 50 123

15 10 9 52 49 120

16 9 9 50 48 116

17 9 8 49 47 113

18 9 8 47 46 110

19 9 8 45 45 107

20 9 8 44 44 105

21 8 8 43 43 102

22 8 7 42 42 99


Intermediate Level Applications

Suppose that we earn simple interest on a bank account. Let’s say that the interest rate is 5% on a principal of $1,000.

a0 = 1000

a1 = 1000 + 1000(.05) = 1050

a2= 1050 + 1000(.05) = 1100

a3 = 1100 + 1000(.05) = 1150

an: 1000, 1050 , 1100 , 1150 , 1200 , …

Compound Interest


Intermediate Level ApplicationsInstead, suppose that we earn 5% interest on a $1,000 principal, compounded annually.

a0 = 1000

a1 = 1000 + 1000(.05) = 1000(1.05) = 1050

a2 = 1050 + 1050(.05) = 1050(1.05) = 1000(1.05)2

a3 = 1000(1.05)3

at = 1000(1.05)t

Compound Interest


Intermediate Level ApplicationsCompound Interest

Most banks and financial institutions offer compound interest which is awarded more frequently than annually, and it is important for students to realize that there is an advantage to getting a fraction of the annual interest more frequently so that more compounding can occur earlier in time.

If yo is the initial principal,

r = the annual percentage rate,

t =the number of years for the money to be invested,

n = the number of times per year that compounding will occur,

yt = yo(1 + )ntn

r



If the number of compoundings is discrete, then this formula is fine. But what if the number of compoundings each year becomes more and more frequent?

Investigate the sequence of (1 + )n as n increases.

1

n

Compound Interest



N ( 1+ ) n

100 2.7048

200 2.7115

1000 2.7169

10,000 2.7181

1,000,000 2.7183

1

n



Note that if n > 1012, the calculator will be subject to some serious roundoff errors. This is because the memory of the calculator only holds about 12 digits, and larger numbers than that overwhelm the capabilities of the machine.

The sequence is (for n = 1, 2, 3, 4, 5…) 2, 2.25, 2.370, 2.441, 2.448, 2.522, 2.545, … , 2.7048 , … 2.7115 , 2.7169 , … , 2.7181 , … 2.7183

e

Compound Interest



There are some wonderful problems for students to solve with interest, and their interest (bad pun…) is piqued with some challenges, such as …

Two people each have $10,000. One invests the money at a 5.1% interest rate, compounded monthly. The other invests at 5% compounded daily. Which investment is better after 8 years? When will they be equal? Which is better after many years?


Intermediate Level ApplicationsLinear and Exponential Functions Compared

Consider two scenarios:

1) Invest $5000 with 5% compound interest earned annually.

2) Invest $5000 and add $500 each year to the account. No interest is earned.

Which investment is better?



The first situation is modeled with an exponential function, since it is geometric sequence.

The second situation is modeled with a linear function, since it is an arithmetic sequence.

Eventually…..if both sequences increase, a geometric sequence will exceed an arithmetic sequence.



Suppose that person has a debt obligation which is subject to a compound annual interest rate of 18% (such as a credit card). The amount owed is $50,000. If the minimum monthly payment is 2.5% of the remaining balance, and the minimum payment is what is made each month, what happens to the debt?

Question: Is a (geometric sequence – arithmetic sequence) a good strategy to pay back a debt? Could it be fine if the minimum payment is high enough?


Intermediate Level ApplicationsAstronomy and Sequences

In the middle of the 19th century, data concerning the distance of the planets in our solar system from the sun indicated that there was a remarkable sequence … with a missing number:

Planet Mercury Venus Earth Mars Jupiter

Dist sun* 36 67.2 92.9 141.6 483.7

A.U.** 0.3875 0.7234 1.0000 1.5242 5.2067

Planet Saturn Uranus Pluto

Dist sun* 890.6 1777 2654.4

A.U.** 9.5867 19.1281 39.3369

* (in millions of miles) ** astronomical units



It seemed that there were two “holes” in the location of the planets, and the location …even the existence (ah, such a word for a mathematician) … of a possible planet was discovered by calculation rather than by observation.

The conclusion was that there was another body pulling Uranus out of the orbit predicted by Bode’s Law, so Adams (England) and Leverrier (France) solved g = to calculate the place where another

planet ought to be found.

1 22

m m

d



Bode’s Law

A B SUM SUM/10

4 4 0.4

4 3 7 0.7

4 6 10 1.0

4 12 16 1.6

4 24 28 2.8

4 48 52 5.2

4 96 100 10.0

4 192 196 19.6

4 384 388 38.8

4 768 772 77.2



Planet A.U. Bode’s Law

A B SUM SUM/10

Mercury 0.3875 4 4 0.4

Venus 0.7234 4 3 7 0.7

Earth 1.0000 4 6 10 1.0

Mars 1.5242 4 12 16 1.6

4 24 28 2.8

Jupiter 5.2067 4 48 52 5.2

Saturn 9.5867 4 96 100 10.0

Uranus 19.1281 4 192 196 19.6

30.1335 4 384 388 38.8

Pluto 39.3369 4 768 772 77.2



On September 23, 1846, astronomers had their telescopes trained on the piece of the night sky where Adams and Leverrier had predicted that a missing planet might be located.

A mere half hour after they began looking, Neptune was observed, only 52 minutes of arc (less than one degree) off from Leverrier’s prediction. It was 2.8 billion miles from earth.

Viva les mathematiques!


Intermediate Level ApplicationsChemistry, Data Analysis and Sequences

Looks for patterns in atomic weight, specific heat or boiling points across rows or down columns.

NB: The TI-84 has a built in periodic table, and there are graphical displays included!


Intermediate Level ApplicationsBiological Growth and Sequences

If a virus grows from a population of 200 at 8 AM to a population of 1000 by noon, how many virus will there be at 4 PM? 6 PM? midnight?

Answer: y0 = 200 (8 AM)

y4 = 1000 (noon, which is 4 hours later)

The sequence is:

t: 0, 4, 8, 12, 16, 20, …

at: 200, 1000, 5000, 25000, 125000, 625000



If we are only interested in the virus counts at whole number of hours, we need the geometric means, and the multiplier becomes

So the sequence is

at = 500 (1.4963t-1)

4 5 1.4953



No wonder healthy people at 8 AM are not feeling well at the ned of a day!


Intermediate Level ApplicationsChemical Half-Life: Radioactivity

The half-life of the chemical element technetium is about 6 hours. This element is used in medicine when tracing body functions, especially renal function or failure in patients receiving chemotherapy. Given the short half-life, what percentage of Tc injected into the body remains after 2 hours? 3 hours? 4 hours?

This is done just as the biological (population) growth, and the hurly percentages can be thought of as a sequence which converges to some value.


Intermediate Level ApplicationsAntibiotic Medications: Sequences and Series

Suppose that an antibiotic medication dissipates in the body so that 20% of the amount currently in the body is gone after 4 hours (or 80% of the medication remains after 4 hours).

A patient is given a 600 mg bolus (a large initial dosage) to begin the treatment. Then the dosage is an additional 100 mg every 4 hours. It is dangerous for the body to have more than 700 mg at any one time, and at least 500 mg is needed to fight the illness (e.g., strep throat).



This is a good example of a problem which can be considered as both a sequence and as a series.

Sequence: a t = amount of medication given at each 4 hour interval

a1 = 600

a2 = 100 + 600(.8) = 580 mg

a3 = 100 + 100(.8) + 600(.82) = 564 mg

a4 = 100 + 100(.8) + 100(.82) + 600(.83) = 551.2 mg

a5 = 100 + 100(.8) + 100(.82) + 100(.83) + 600(.84) = 540.96 mg



The medication after the bolus forms a geometric sequence which decreases to zero, and the repeated medications form a geometric series:

sum = 100 + 100(.8) + 100(.82) + 100(.83) + …

= = = 500 1

1

a

r100

1 .8

The combined dosages (which are a series) form a sequence which needs to stay between the effective and the dangerous drug levels.

(What happens to the original bolus?)


Intermediate Level ApplicationsCooling of Liquids

A hot cup of coffee ( of cocoa, tea, …) fresh from the coffeepot has a temperature of 140o F.

a) How does it cool?

b) This can be simulated with a CBL and TI-83/84.

c) Use appropriate data analysis and regressions.



Time (min) Temp1 Temp2 Temp3 Temp4

0 140 140 140 140

1 135 126 133 130

2 130 113.4 126.7 121

3 125 102.1 121 113

4 120 91.9 115.9 106

5 115 82.67 111.3 100

10 90 48.8 94.4 85

20 65 17 78.5 ??

Which sequence of

temperatures makes the

most sense?

How are each of the sequences calculated?



It seems to be good, authentic mathematics and science to guess which of the sequences is most reasonable, and then try to fit a function to that sequence.

Following such intuition with a data collection with a CBL on cooling water will give data to verify or refute the earlier guess.

Newton’s Law of Cooling:

= k(T-Tambient) or T-Ta = (T0-Ta)e -kt( )ambientd T T

dt


Intermediate Level ApplicationsEconomics

When a yearbook is printed, suppose it costs $9000 to print one copy, because of the set-up costs for the press, type-setting, importing photographs, binding, cover set-up, and artwork. It costs as additional $8 for each book, since the press is already set up, and only paper, binding, and some ink are needed for the second copy.

1. What is the cost of 5 books? 10 books? 100 books? n books?

2. What is the average cost of n books?

3. What is the difference in average costs for printing n to (n+1) books for various values of n?


Intermediate Level ApplicationsEconomics

As a sequence or series problem,

b1 = 9000 b1 average = 9000/1 = 9000

b2 = 9000 + 8 b2 average = 9008/2 = 4504

b3 = 9000 + 8 + 8 b3 average = 9016 / 3 = 3005.33

bn = 9000 + 8(n-1) bn average = (8992 + 8n) / n

= 8992/n + 8If 500 yearbooks are ordered, it costs $12,992 to print them, and the average cost is $25.98

This can be taught as a sequence problem or as a rational function problem.

Advanced Sequences and Series 111

Advanced LevelCalculus Examples, especially AP Calculus

Newton’s Method uses the definition of derivative to provide a method to locate the roots of a function. (It differs from the algorithm ROOT FINDER in the TI-83/84 calculators which uses the IVT)

x n+1 = xn –

This is an iterative algorithm, where the results (output) of each stage become the input of the next stage. If we look at each xn and its subsequent xn+1, then the fraction which is subtracted can be considered as the “correction factor”, which (hopefully) sends us closer, via a sequence, to the exact location of the root of a function.

( )

'( )n

n

f x

f x



An example of Newton’s method:

Suppose we want to approximate

This is a root of f(x) = x2 – 3

The sequence of values from Newton’s Method looks like:

x0 = 1 (our choice for a “guess”)

3

The sequence seems to converge.



An example of Newton’s method:Suppose we want to approximate the roots of

f(x) = x2 +

The sequence of values from Newton’s Method looks like:x0 = 1 (our choice for a “guess”)

3 x

This time, there is no convergence, and we cannot locate a root.



Riemann sums are the basis for evaluating the area under a function, as sums of the areas of rectangles are used to approximate the exact area.

It is probably a good idea to mention the words sequence and series in the explanation for the strategy. After all, the “C” part of the BC Calculus concerns the ideas of series and convergence, but the ideas of the convergence of sequences and series can appear very early in the “A” part of the differential calculus when limits are discussed and when early ideas about areas under functions are introduced.



10

8

6

4

2

-10 -5 5 10

f x = x2+1

There is a sequence of the areas of each rectangle, and there is a sequence of the partial sums of the rectangles. Convergence of each of these is an important idea.



10

8

6

4

2

-10 -5 5 10

f x = x2+1

…and as the number of partitions goes from 4 to 8 to 16 to …, there is a

sequence of estimates on the area, and the idea for calculus students is to believe that the sequence of series converges…to the exact area.



We usually consider the Trapezoid Rule and Simpson’s Rule as series, but if we repeat them with more and more partitions, then sequence of the series should converge.

Trap =

Simpson =

0 1 2 1

1( ) ( 2 2 ... 2 )

2

b

n n

a

f x dx h y y y y y

0 1 2 3 4 2 1

1( ) ( 4 2 4 2 ... 4 2 )

3

b

n n n

a

f x dx h y y y y y y y y n = an even number of partitions required for Simpson’s rule



Evaluate using different algorithms. 4

2

0

x dxUpper Lower Trapezoids Simpson

n=4 30 14 22 21.3333

N = 8 25.5 17.5 21.5 21.3333

n = 20 22.96 19.76 21.36 21.3333

n = 100 21.9776 20.6976 21.3376 21.3333

n = 50 21.56544 21.0144 21.3344 21.3333

n = 1000 21.365 21.301344 21.33334 21.3333



Euler’s method is an iterative algorithm to give approximate solutions to differential equations. It is really just a linearization method that is used repeatedly to give a sequence of points which serve as a numerical function.

1 *n n

dyy y x

dx



Example: Solve with the initial condition (1, 2)

Use = 0.1

answer: The first point on the solution is (1,1) because x0 = 2 and y0 = 1

= 2 + [(1)(22)] (0.1)

= 2.4 the second point is (1.1, 2.4)

2dyxy

dx

x

1 *n n

dyy y x

dx



So x1 = 1.1 and y1 = 2.4

= 2.4 + [(1.1)(2.42)] (0.1)= 3.0336 the third point is (1.2, 3.0336)

We continue the process, generating a sequence of approximate solutions to the differential equation. (If is smaller, then the theory says that the sequence should more closely match the function which is the solution to the differential equation.)

1 *n n

dyy y x

dx

x



The exact solution, using separable differential equation methods, is

It is not always the case that an exact solution can be found, and those are the examples for which the approximate solutions algorithms are important.

2

2

2y

x

There are also some algorithms which provide more accurate approximations. Two of them are called the Improved Euler method and the Runge-Kutta Method.



A summary of the results of these algorithms is:

X Y

(exact)

Y

(Euler)

Y

(ImprovedEuler)

Y

(Runge-Kutta)

1 2 2 2 2

1.1 2.5316 2.4 2.5168 2.5316

1.2 3.5714 3.0336 3.4848 3.5706

1.3 6.4516 4.1379 5.80101 6.4304

perfect! good better best



Maclaurin and Taylor polynomials are a series of polynomial (power) terms, and they are typically taught near the end of a BC Calculus course. A suggestion is to introduce them much earlier in the course, since students only need to be able to do derivatives to calculate these series. Then the approximation methods that they provide with “polynomials simulating other function” can be used, for example, when an indefinite or definite integral is to be done, and students have not yet learned the antiderivative of that function.

We want to convince them that the polynomial (or power) series is a sequence of series that converges.



A couple of ideas to emphasize the ideas of sequences of series with Maclaurin and Taylor polynomials.

a. Show simultaneously the graphs of

y = sin x

y =

3 5 7

...3! 5! 7!

x x xx

Put increasingly more terms of the series in the calculator to see how the original function and its Maclaurin series match.



b. Show, graphically, the limited convergence of

y = ln (x + 1)

y =

2 3 4

...2 3 4

x x xx

This will provide a good foundation for understanding the “convergence tests” and “intervals of convergence” ideas which follow.



c. Evaluate using a series for y =

centered at x = 9.

Show that a Taylor series is easier to evaluate (“easier” = “uses

only simple arithmetic”) and can almost be done without a

calculator at all.

9.08 x



James Gregory’s method for estimating (1671)

Since = tan -1 (1) which equals the value of

then a Maclaurin series for the integrand can be antidifferentiated and an approximate value can be done with ordinary arithmetic.

4

1

20

1

1dx

x



The sequence of operations that is useful here is:

1.2 3 4

2 4 6 82

11 ...

1

11 ...

1

x x x xx

x x x xx

This last series is accomplished by replacing the x’s by x2’s in the first series. This is a very helpful (and yet unique) feature of Maclaurin series!



2. Then

= 0.8349206349 (ok, so I did use a calculator to do some of this!!)

Then = 4 (0.8349206349) = 3.33968254

113 5 7

2 00

1 1 1 1...

1 3 5 7dx x x x x

x



3. We can update what James Gregory did, using technology to see whether his series converges.

The antiderivative series can be written as 1

1

1( 1) *

(2 1)n

n n

On a TI-83/84,

put the counters n is L1 (as far as you want to go)

the terms of the sequence in L2 as (-1)^(L1+1) /(2L1 -1)

the accumulated sum in to L3 as cumSum(L2)

The series does not converge very quickly, so it is not useful, but it is a valuable method to teach.

132

Sequences and SeriesFrom Simple Patterns to Elegant and Profound Mathematics

Mathematics is all about expressing patterns, numerically and graphically.

Patterns can indicate some interesting, usual, unusual, and sometimes complicated simulations of real phenomena.

So sequences and series ought to be as much a part of our mathematical language as functions, formulas, equations, expressions, and shapes.

1 Sequences and Series From Simple Patterns to Elegant and Profound Mathematics David W. Stephens The Bryn Mawr School Baltimore, Maryland PCTM – 28 October.

Documents

early sequences

patterns of sequences

advanced sequences

series algebra

teaching sequences

intermediate sequences

geometric sequences

kinds of sequences