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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

Contact Information

- stephensd_at_brynmawrschool.org
- The post office mailing address is
- David W. Stephens
- 109 W. Melrose Avenue
- Baltimore, MD 21210
- 410-323-8800
- 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

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)

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.

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

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)

Advanced Sequences and Series (AP Calculus, esp.

BC Calculus)

- 1. Newtons method for locating roots
- 2. Riemann sums
- 3. Trapezoid rule and Simpsons rule
- 4. Eulers method for differential equations
- 5. Power series (Maclaurin and Taylor series

polynomials) - 6. Convergence tests for series

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 , ...

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

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

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

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

Activity 8

- For the series,
- sn 5 10 20 40 320
- calculate s28

Activity 9

Activity 10

Data Analysis Building Functions from Data

Data Analysis Building Functions from Data

Data Analysis Building Functions from Data

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.

GeometryA Modeling Application

Handshake Problem

GeometryA Modeling Application

- n number of people
- h(n) number of handshakes
- n 1 2 3 4 5 6 7
- h(n) 0 1 3 6 10 15 21

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

GeometryA Derivation of

s length of one side p perimeter of inscribed

polygon

s p 4 5.657

s p 3 5.196

GeometryA Derivation of

s 1 p 6

s 2 sin(36) 1.176 p 5(1.176) 5.878

GeometryA Derivation of

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.

GeometryA Derivation of

The central angle for each side is

Each half-side has length equal to the sine of

one-half the central angle.

GeometryA Derivation of

Here are the perimeters of the polygons from the

TI-83 as a list (L2) Note Ignore L3.

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)

Data AnalysisBuilding Functions from Data

- 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 ys, then we see that this

sequence is 2 more than a geometric sequence. - So y 3x 2

Data AnalysisBuilding Functions from Data

- 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 xs do not have

to be consecutive. - We have to find the r value as if we are

calculating geometric means

Data AnalysisBuilding Functions from Data

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

48/6. - So r 2 , and y 3 2x

Data AnalysisBuilding Functions from Data

- 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

An Historical Diversion

Lets 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 Lets 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

An Historical Diversion

- 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

An Historical Diversion

- 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.

Function Transformations using Sequences

- If 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

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

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)

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 af(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

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

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

g(x) 2f(x 2)

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.

Infinite Sequences, Series and Convergence

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?

Infinite Sequences, Series and Convergence

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.

Infinite Sequences, Series and Convergence

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 .8 Repeatedly

hit enter to generate the sequence.

Infinite Sequences, Series and Convergence

Suppose 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?

Infinite Sequences, Series and Convergence

Suppose in Where does in go as n gets

large?

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/ltOption 5gt which will give

seq( The inputs required for seq(

are seq(expression, variable, begin, end

increment) Example an 2n1 ? 1, 3, 5,

7, 9,

Intermediate Level ApplicationsSequence and

Series on the Calculator

Example an 2n1 ? 1, 3, 5, 7, 9,

Notice that the name of the variable does not

matter, as long as it is specified.

Intermediate Level ApplicationsSequence and

Series on the Calculator

If the series is desired, the sum( function

is used. Example an 2n1 ? 1, 3, 5, 7,

9 sn sum(an) ? 13579 25

Sum( is found in LIST (2nd STAT)/MATH/ltOption

5gt

Intermediate Level ApplicationsSequence and

Series on the Calculator

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/ltOption 6gt Example an

2n1 ? 1, 3, 5, 7, 9 cumSum(an) ? 1, 4, 9,

16, 25

Intermediate Level ApplicationsSequence and

Series on the Calculator

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

or cumSum(seq(2N1, N, 0, 4))

If a list is already in the calculator, perhaps

in L1, then cumSum(L1) or sum(L1) will give

series results.

Intermediate Level ApplicationsSequence Mode on

the Calculator

First term

First term value

Recursive function

Intermediate Level ApplicationsSequence Mode on

the Calculator

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.

Intermediate Level ApplicationsSequence Mode 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

Intermediate Level ApplicationsSequence Mode on

the Calculator

Go to the 2nd ZOOM Format key, and make sure

that TIME is selected at the top. Hit GRAPH.

u(n)0.8u(n-1)

This is a scatterplot of the (n, an)

Intermediate Level ApplicationsSequence Mode on

the Calculator

Instead, select 2nd Zoom Format and choose

Web. Set the Window to 0 lt xlt 105 and 0 lt y lt

105 Hit Graph and Trace. Hit the right arrow to

iterate the web.

Intermediate Level ApplicationsSequence Mode on

the Calculator

Lets 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

Intermediate Level ApplicationsSequence Mode on

the Calculator

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(10.05 -.001Wn-1)

rabbits Wn Wn-1(10.0002Rn-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.

Intermediate Level ApplicationsSequence Mode on

the Calculator

Intermediate Level ApplicationsSequence Mode on

the Calculator

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.

Intermediate Level ApplicationsSequence Mode on

the Calculator

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.

Intermediate Level ApplicationsSequence Mode on

the Calculator

Experimentation with the data suggests that the

new WINDOW be XMin 80 XMax 250 Xscl

50 YMin 0 YMax 100 YScl 10

Infinite Sequences, Series and Convergence

When series go on forever, we call them infinite

series. Lets look at arithmetic series

first. Sn 4 7 10 13 What is the

sum s100? s1000 ? s 1,000,000 ?

Infinite Sequences, Series and Convergence

- Compare the results for each of these arithmetic

series - Sn 4 7 10 13
- Sn 1 1.1 1.2 1.3 1.4
- Sn 5 5.001 5.000001
- Sn 4 3.5 3 2.5 2
- Conclusion ?

Infinite Sequences, Series and Convergence

- Moving onto geometric series, consider the

behavior of these sums by taking the number of

terms to be higher and higher. - sn 2 4 8 16
- sn
- s10 s100 s1000

Infinite Sequences, Series and Convergence

- sn 2 2(1.02) 2(1.02)2 2(1.02)3
- sn2 2(0.98) 2(.98)2 2(.98)3
- sn 1 3 9 27 81
- sn 1
- sn 1

Infinite Sequences, Series and Convergence

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 lt r

lt 1 or r lt 1

Infinite Sequences, Series and Convergence

Looking at the sequences graphically makes some

strong connections with algebra, and the visual

impact helps with understanding about convergence

and divergence. Lets 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

Examples an bn

cn dn

Graphs of Sequences and Series

Graphs of Sequences and Series

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.

Intermediate Level ApplicationsDeer Populations

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

Intermediate Level ApplicationsDeer Populations

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. Lets 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 Applications

Instead, 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 )nt

Intermediate Level Applications

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.

Compound Interest

Intermediate Level ApplicationsCompound Interest

N ( 1 ) n

100 2.7048

200 2.7115

1000 2.7169

10,000 2.7181

1,000,000 2.7183

Intermediate Level Applications

Note that if n gt 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

Intermediate Level ApplicationsCompound 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
- Invest 5000 with 5 compound interest earned

annually. - Invest 5000 and add 500 each year to the

account. No interest is earned. - Which investment is better?

Intermediate Level ApplicationsLinear and

Exponential Functions Compared

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.

Intermediate Level ApplicationsLinear and

Exponential Functions Compared

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

Intermediate Level ApplicationsAstronomy and

Sequences

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 Bodes Law, so Adams (England) and

Leverrier (France) solved g to

calculate the place where another planet ought

to be found.

Intermediate Level ApplicationsAstronomy and

Sequences

Bodes 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

Intermediate Level ApplicationsAstronomy and

Sequences

Planet A.U. Bodes 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

Intermediate Level ApplicationsAstronomy and

Sequences

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 Leverriers 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

Intermediate Level ApplicationsBiological Growth

and Sequences

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)

Intermediate Level ApplicationsBiological Growth

and Sequences

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).

Intermediate Level ApplicationsAntibiotic

Medications Sequences and Series

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

Intermediate Level ApplicationsAntibiotic

Medications Sequences and Series

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

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.

Intermediate Level ApplicationsCooling of Liquids

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?

Intermediate Level ApplicationsCooling of Liquids

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.

Newtons Law of Cooling k(T-Tambient)

or T-Ta (T0-Ta)e -kt

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. - What is the cost of 5 books? 10 books? 100 books?

n books? - What is the average cost of n books?
- What is the difference in average costs for

printing n to (n1) 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

8

If 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 LevelCalculus Examples, especially AP

Calculus

Newtons 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 n1 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 xn1,

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.

Advanced LevelCalculus Examples, especially AP

Calculus

An example of Newtons method Suppose we want

to approximate This is a root of f(x) x2

3 The sequence of values from Newtons Method

looks like x0 1 (our choice for a

guess)

The sequence seems to converge.

Advanced LevelCalculus Examples, especially AP

Calculus

An example of Newtons method Suppose we want

to approximate the roots of f(x) x2

The sequence of values from Newtons Method

looks like x0 1 (our choice for a guess)

This time, there is no convergence, and we cannot

locate a root.

Advanced LevelCalculus Examples, especially AP

Calculus

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.

Advanced LevelCalculus Examples, especially AP

Calculus

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.

Advanced LevelCalculus Examples, especially AP

Calculus

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

convergesto the exact area.

Advanced LevelCalculus Examples, especially AP

Calculus

We usually consider the Trapezoid Rule and

Simpsons Rule as series, but if we repeat them

with more and more partitions, then sequence of

the series should converge. Trap Simpson

n an even number of partitions required for

Simpsons rule

Advanced LevelCalculus Examples, especially AP

Calculus

Evaluate using different

algorithms.

Upper Lower Trapezoids Simpson

n4 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

Advanced LevelCalculus Examples, especially AP

Calculus

Eulers 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.

Advanced LevelCalculus Examples, especially AP

Calculus

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)

Advanced LevelCalculus Examples, especially AP

Calculus

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.)

Advanced LevelCalculus Examples, especially AP

Calculus

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.

There are also some algorithms which provide more

accurate approximations. Two of them are called

the Improved Euler method and the Runge-Kutta

Method.

Advanced LevelCalculus Examples, especially AP

Calculus

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

Advanced LevelCalculus Examples, especially AP

Calculus

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.

Advanced LevelCalculus Examples, especially AP

Calculus

- A couple of ideas to emphasize the ideas of

sequences of series with Maclaurin and Taylor

polynomials. - Show simultaneously the graphs of
- y sin x
- y

Put increasingly more terms of the series in the

calculator to see how the original function and

its Maclaurin series match.

Advanced LevelCalculus Examples, especially AP

Calculus

- Show, graphically, the limited convergence of
- y ln (x 1)
- y

This will provide a good foundation for

understanding the convergence tests and

intervals of convergence ideas which follow.

Advanced LevelCalculus Examples, especially AP

Calculus

- 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.

Advanced LevelCalculus Examples, especially AP

Calculus

- James Gregorys 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.

Advanced LevelCalculus Examples, especially AP

Calculus

The sequence of operations that is useful here

is 1.

This last series is accomplished by replacing the

xs by x2s in the first series. This is a very

helpful (and yet unique) feature of Maclaurin

series!

Advanced LevelCalculus Examples, especially AP

Calculus

- Then
- 0.8349206349 (ok, so I did use a calculator

to do some of this!!) - Then 4 (0.8349206349) 3.33968254

Advanced LevelCalculus Examples, especially AP

Calculus

- We can update what James Gregory did, using

technology to see whether his series converges. - The antiderivative series can be written

as

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)(L11) /(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.

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.

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