The Hindu – Arabic Series

At first glance, the topic of series seems to be something encountered late in the year of a BC Calculus course. But everyone uses series any time they use numbers, which is to say very often. Let’s look at this particularly important series.  

The way numerals were written way back when was clumsy. If you don’t believe me, try multiplying or dividing with Roman Numerals. Around AD 1200, Leonardo of Pisa introduce the Hindu-Arabic system for writing numbers to Europe. He learned this system during his travels in the Middle East. Leonardo is also known as Fibonacci although he was not given that name until the 1500s. While he is better known for his famous sequence, I think improving the way numbers are written is a much more important contribution to mathematics (even though it would have caught on eventually). This is the system used world-wide today.

A page of Fibonacci’s Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals (black) and the value in Hindu-Arabic numerals (red). (Some numerals are slightly different than those in use today.)

Hindu – Arabic notation is a shorthand for a series, sometimes finite, often infinite.  It is a sophisticated idea when thought of in modern terms. This “new” system is a place-value system: the value of each digit depends on its position relative to the decimal point, defined by a sequence. For example,

\displaystyle \begin{array}{l}456.789=4(100)+5(10)+6+7(\tfrac{1}{{10}})+8(\tfrac{1}{{100}})+9(\tfrac{1}{{1000}})\\\end{array}

This notation has advantages over other methods of denoting numbers. The decimal representation of a number is unique (well almost, as we’ll see below), whereas every Rational number may be written as many different fractions. Certainly, makes computation easier. It also makes finding approximations and arranging numbers in order easier than writing them as fractions.

But other things also happen.

The Hindu-Arabic decimal system revealed that all Rational numbers written in this notation are repeating decimals. A repeating decimal is an expression containing a string of one or more digits that repeated forever. For example, 1/3 = 0.333333… with the “3” repeating forever and \displaystyle \frac{{241}}{{55}}=4.38181818... with “18” string repeating forever. (Some numbers repeat zeros forever; they are a special case called terminating decimals.)

The decimal form of a fraction may be found by using the division algorithm. Since only those numbers less than the divisor may appear as “remainders,” eventually one of them will appear again after which the succeeding digits will repeat.

Conversely, any repeating or terminating decimal can be written as a quotient of integers. This example shows the procedure.                                                              

Let \displaystyle n=4.3818181818...

Then \displaystyle 100n=438.18181818...

Subtracting the first from the second

\displaystyle 99n=433.8

\displaystyle n=\frac{{433.8}}{{99}}=\frac{{4338}}{{990}}=\frac{{241}}{{55}}

So, all the Rational numbers can be written as a repeating or terminating decimal, and conversely all repeating or terminating decimals are Rational numbers. Numbers that cannot be written as repeating or terminating decimals are exactly the Irrational numbers.  

An Irrational Number – the Diagonals of a Square

The length of the diagonals of a square is a non-repeating decimal. That is, the length must be expressed as an infinitely long decimal that contains no string of digits that repeats. \displaystyle \sqrt{2} is an Irrational Number and there are a lot of others like it!

By the Pythagorean Theorem the diagonals of a square with sides of one unit have a length denoted by \displaystyle \sqrt{2} – the number whose square is 2. In a previous post, I showed a way, one of several, to find closer and closer decimal approximations to \displaystyle \sqrt{2}. The table below shows the results.


n = decimal places
Ln < \displaystyle \sqrt{2}Gn > \displaystyle \sqrt{2}
012
11.41.5
21.411.42
31.4141.415
41.41421.4143
51.414211.41422
61.4142131.414214
71.41421351.4142136
81.414213561.41423157
91.4142135621.414213563
101.41421356231.4142135624
111.414213562371.41421356238
121.4142135623731.414213562374
131.41421356237301.4142135623731
141.414213562373091.41421356237310
151.4142135623730951.414213562373096
161.41421356237309501.4142135623730951
171.414213562373095041.41421356237309505
   
Each number in list Ln was produced by taking the preceding number and affixing the digits 0, 1, 2, … , 9 to it, squaring that number, and finding the largest whose square was less than 2. The Gn is the next number, the smallest number with a square greater than 2. The numbers in Ln have squares less than 2; the numbers in Gn have squares greater than 2.

The Ln list is a sequence of numbers that has two important properties easily seen from how it was developed: (1) it is non-decreasing – each number is greater than or occasionally equal to the preceding number because each time we append an extra digit we get a greater number, and (2) the list is bounded above – the numbers never exceed 100, or 15, or \displaystyle \pi , or 2, or in fact any number from the Gn list. The smallest number they never exceed is \displaystyle \sqrt{2}. We know this because this is how the list was developed.

The number that’s between the two lists is the number we’re looking for is \displaystyle \sqrt{2}, but we can never find an “exact” decimal representation. The two lists give better and better approximations to \displaystyle \sqrt{2}. They close in on it. But neither gets there.

So, how do we know that number exists?

The Axiom of Completeness

All the decimal numbers, the Rational Numbers, and the Irrational Numbers (and no others), make up a set called the Real Numbers.

If this list above can be continued forever, it will never get to \displaystyle \sqrt{2}. To handle this kind of situation a new rule (called an axiom) was imposed.

The Axiom of Completeness: Every non-decreasing sequence of Real numbers that is bounded above converges to – gets closer and closer to – its least upper bound.

Since this is an axiom, it is not proved; it is just accepted as fact.

The axiom says that even though there is no decimal to represent it, the number nevertheless exists.

Someone made the axiom up. This is very different than observing and naming a property of numbers like the commutative property or the associative property. It doesn’t have to be true, but it seems very reasonable, and no one has ever found a counterexample. [1]

In the example, the least upper bound is \displaystyle \sqrt{2}. How do we know that? Because that’s what we made the sequence to do. Any other method of “finding” \displaystyle \sqrt{2}, and there are many, gives us, not just the same kind of thing, but the exact same list! Creepy, isn’t it?

Thus, \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{L}_{n}}} \right\}=\sqrt{2}

Not only are all Irrational Numbers handled the same way, but all repeating decimals are also. They never repeat digits and, however you find their decimal approximations, the same thing happens: you have a non-increasing sequence of numbers that is bounded above and, by the Axiom of Completeness, converges to the fraction.

I mentioned above that the Hindu-Arabic system gives a unique expression for every number. Not quite. Consider 1/3 = 0.3333333 ….  Since three times one-third is one, it must be that 3 times this decimal which is 0.9999999…. = 1.

At first, I disliked decimals because they were not “exact.” I got over that. For the cost of the Axiom of Completeness, we have a system for writing numbers that makes computation easy (if, often, only to a very good approximation). It’s worth the cost.

But does \displaystyle \sqrt{2} exist? Is it really there?


[1] As a corollary to the axiom, there is a theorem that says a non-increasing sequence that is bounded below, such as Gn, converges to its greatest lower bound, again \displaystyle \sqrt{2}.

Sequences and Series (Type 10)

AP Questions Type 10: Sequences and Series (BC Only)

The last BC question on the exams usually concerns sequences and series. The question may ask students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in multiple-choice questions. Questions about the convergence of sequences may appear as multiple-choice questions. With about 8 multiple-choice questions and a full free-response question this is one of the major topics on the BC exams.

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a convergence test chart students should be familiar with; this list is also on the resource page.

Students should be familiar with and able to write a few terms and the general term of a Taylor or Maclaurin series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it, or integrating it.

The general form of a Taylor series is \displaystyle \sum\limits_{{n=0}}^{\infty }{{\frac{{{{f}^{{\left( n \right)}}}\left( a \right)}}{{n!}}{{{\left( {x-a} \right)}}^{n}}}}; if a = 0, the series is called a Maclaurin series.

What Students Should be Able to Do 

  • Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.  and the posts “What Convergence Test Should I use?” Part 1 and Part 2. In 2022 BC 6 (a) students were asked to state the condition (hypotheses) of the convergence test they were asked to use.
  • Understand absolute and conditional convergence. If the series of the absolute values of the terms of a series converges, then the original series is said to be absolutely convergent (or converges absolutely). If a series is absolutely convergent, then it is convergent. If the series of absolute values diverges, then the original series may or may not converge; if it converges it is said to be conditionally convergent.
  • Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
  • Distinguish between the Taylor series for a function and the function. DO NOT say that the Taylor polynomial is equal to the function (this will lose a point); say it is approximately equal.
  • Determine a specific coefficient without writing all the previous coefficients.
  • Write a series by substituting into a known series, by differentiating or integrating a known series, or by some other algebraic manipulation of a series.
  • Know (from memory) the Maclaurin series for sin(x), cos(x), ex and \displaystyle \frac{1}{{1-x}}and be able to find other series by substituting into one of these.
  • Find the radius and interval of convergence. This is usually done by using the Ratio test to find the radius and then checking the endpoints. for a geometric series, the interval of convergences is the open interval \displaystyle -1<r<1 where r is the common ration of the series.
  • Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, \displaystyle {{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}}. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
  • Be familiar with the harmonic and alternating harmonic series. These are often useful series for comparison.
  • Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
  • Determine the error bound for a convergent series (Alternating Series Error Bound or Lagrange error bound). See my posts on Error Bounds and the Lagrange Highway
  • Use the coefficients (the derivatives) to determine information about the function (e.g., extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the free-response section is usually quite straightforward. Topics and convergence tests may appear on the multiple-choice section. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked, how it is a sked, and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series or any other topic you are interested in.

Free-response questions:

  • 2004 BC 6 (An alternate approach, not tried by anyone, is to start with \displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin \left( {5x} \right)\cos \left( {\tfrac{\pi }{4}} \right)+\cos \left( {5x} \right)\sin \left( {\tfrac{\pi }{4}} \right)). See Good Question 16
  • 2011 BC 6 (Lagrange error bound)
  • 2016 BC 6
  • 2017 BC 6
  • 2019 BC 6
  • 2021 BC 5 (a)
  • 2021 BC 6 – note that in (a) students were required to state the conditions of the convergence test they were asked to use.
  • 2022 BC 6 – Ratio test, interval of conversion with endpoint analysis, Alternating series error bound, series for derivative, geometric series.

Multiple-choice questions from non-secure exams:

  • 2008 BC 4, 12, 16, 20, 23, 79, 82, 84
  • 2012 BC 5, 9, 13, 17, 22, 27, 79, 90

These questions come from Unit 10 of the CED.


Revised March 12, 2021, April 12, 16, and May 14, 2022


Sequences

Here is a list of past posts on the topics of sequences and series that I hope you find interesting and useful. The first two are suitable for precalculus students.

The first uses sequences and series for a very practical aim that affects almost everyone sometime in their life: paying off a loan. The next gives a good, and I hope, understandable explanation of what an irrational number is.

Amortization. When you have a mortgage on your home or your car, you make the same payment every month. Part of the money pays the interest on the outstanding balance for the last month; the rest pays down the principal so there is less to pay interest on next month. How is the payment computed?

A Lesson on Sequences. What is the square root of two? Really, what is it? This post is an outline of a lesson finding a sequence of numbers that converges to a specific number known in advance and by doing so defines that number.

The next three posts deal with convergences tests and are of interest to BC students at this time of year.

Reference Chart. An outline of the various convergence tests, and their hypotheses (when you can use them).

These two posts answer the question in their titles:

Which Convergence Test Should I Use? Part 1: Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2: Specific hints and a discussion of the usefulness of absolute convergence

Infinite Sequences and Series – Unit 10

Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topics 10.1 – 10.2

Topic 10.1: Defining Convergent and Divergent Series.

Topic 10. 2: Working with Geometric Series. Including the formula for the sum of a convergent geometric series.

Topics 10.3 – 10.9 Convergence Tests

The tests listed below are tested on the BC Calculus exam. Other methods are not tested. However, teachers may include additional methods.

Topic 10.3: The nth Term Test for Divergence.

Topic 10.4: Integral Test for Convergence. See Good Question 14

Topic 10.5: Harmonic Series and p-Series. Harmonic series and alternating harmonic series, p-series.

Topic 10.6: Comparison Tests for Convergence. Comparison test and the Limit Comparison Test

Topic 10.7: Alternating Series Test for Convergence.

Topic 10.8: Ratio Test for Convergence.

Topic 10.9: Determining Absolute and Conditional Convergence. Absolute convergence implies conditional convergence.

Topics 10.10 – 10.12 Taylor Series and Error Bounds

Topic 10.10: Alternating Series Error Bound.

Topic 10.11: Finding Taylor Polynomial Approximations of a Function.

Topic 10.12: Lagrange Error Bound.

Topics 10.13 – 10.15 Power Series

Topic 10.13: Radius and Interval of Convergence of a Power Series. The Ratio Test is used almost exclusively to find the radius of convergence. Term-by-term differentiation and integration of a power series gives a series with the same center and radius of convergence. The interval may be different at the endpoints.

Topic 10.14: Finding the Taylor and Maclaurin Series of a Function. Students should memorize the Maclaurin series for \displaystyle \frac{1}{{1-x}}, sin(x), cos(x), and ex.

Topic 10.15: Representing Functions as Power Series. Finding the power series of a function by, differentiation, integration, algebraic processes, substitution, or properties of geometric series.


Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.


Previous posts on these topics:

Before sequences

Amortization Using finite series to find your mortgage payment. (Suitable for pre-calculus as well as calculus)

A Lesson on Sequences An investigation, which could be used as early as Algebra 1, showing how irrational numbers are the limit of a sequence of approximations. Also, an introduction to the Completeness Axiom. 

Everyday Series

Convergence Tests

Reference Chart

Which Convergence Test Should I Use? Part 1 Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2 Specific hints and a discussion of the usefulness of absolute convergence

Good Question 14 on the Integral Test

Sequences and Series

Graphing Taylor Polynomials Graphing calculator hints

Introducing Power Series 1

Introducing Power Series 2

Introducing Power Series 3

New Series from Old 1 substitution (Be sure to look at example 3)

New Series from Old 2 Differentiation

New Series from Old 3 Series for rational functions using long division and geometric series

Geometric Series – Far Out An instructive “mistake.”

A Curiosity An unusual Maclaurin Series

Synthetic Summer Fun Synthetic division and calculus including finding the (finite)Taylor series of a polynomial.

Error Bounds

Error Bounds Error bounds in general and the alternating Series error bound, and the Lagrange error bound

The Lagrange Highway The Lagrange error bound. 

What’s the “Best” Error Bound?

Review Notes

Type 10: Sequences and Series Questions


 

 

 

 

 

Sequences and Series (Type 10)

AP Questions Type 10:  Sequences and Series (BC Only)

The last BC question on the exams usually concerns sequences and series. The question may ask students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in multiple-choice questions. Questions about the convergence of sequences may appear as multiple-choice questions. With about 8 multiple-choice questions and a full free-response question this is one of the largest topics on the BC exams.

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a convergence test chart students should be familiar with; this list is also on the resource page.

Students should be familiar with and able to write several terms and the general term of a Taylor or Maclaurin series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it, or integrating it.

The general form of a Taylor series is \displaystyle \sum\limits_{n=0}^{\infty }{\frac{{{f}^{\left( n \right)}}\left( a \right)}{n!}{{\left( x-a \right)}^{n}}}; if a = 0, the series is called a Maclaurin series.

What Students Should be Able to Do 

  • Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.  and the posts “What Convergence Test Should I use?” Part 1 and Part 2
  • Understand absolute and conditional convergence. If the series of the absolute values of the terms of a series converges, then the original series is said to absolutely convergent (or converges absolutely). If a series is absolutely convergent, then it is convergent. If the series of absolute values diverges, then the original series may or may not converge; if it converges it is said to be conditionally convergent.
  • Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
  • Distinguish between the Taylor series for a function and the function. DO NOT say that the Taylor polynomial is equal to the function (this will lose a point); say it is approximately equal.
  • Determine a specific coefficient without writing all the previous coefficients.
  • Write a series by substituting into a known series, by differentiating or integrating a known series, or by some other algebraic manipulation of a series.
  • Know (from memory) the Maclaurin series for sin(x), cos(x), ex and \displaystyle \tfrac{1}{1-x} and be able to find other series by substituting into one of these.
  • Find the radius and interval of convergence. This is usually done by using the Ratio test to find the radius and then checking the endpoints.
  • Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, \displaystyle {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
  • Be familiar with the harmonic and alternating harmonic series. These are often useful series for comparison.
  • Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
  • Determine the error bound for a convergent series (Alternating Series Error Bound and Lagrange error bound). See my posts on Error Bounds and the Lagrange Highway
  • Use the coefficients (the derivatives) to determine information about the function (e.g. extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the free-response section is usually quite straightforward. Topics and convergence test may appear on the multiple-choice section. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series or any other topic you are interested in.

Free-response questions:

  • 2004 BC 6 (An alternate approach, not tried by anyone, is to start with \displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin (5x)\cos \left( {\tfrac{\pi }{4}} \right)+\cos (5x)\sin \left( {\tfrac{\pi }{4}} \right))
  • 2011 BC 6 (Lagrange error bound)
  • 2016 BC 6
  • 2017 BC 6
  • 2019 BC 6

Multiple-choice questions from non-secure exams:

  • 2008 BC 4, 12, 16, 20, 23, 79, 82, 84
  • 2012 BC 5, 9, 13, 17, 22, 27, 79, 90,

These question come from Unit 10 of the  2019 CED.


 

 

 

 

Revised March 12, 2021


 

 

 

2019 CED Unit 10: Infinite Sequences and Series

Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

Topics 10.1 – 10.2


Timing

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.


Previous posts on these topics :

Introducing Power Series 1

A Lesson on Sequences

This blog post describes a lesson that investigates some ideas about sequences that are not stressed in the AP Calculus curriculum. The lesson could be an introduction to sequences. I think the lesson would work in an Algebra I course and is certainly suitable for a pre-calculus course. The investigation is of irrational numbers and their decimal representation. The successive decimal approximations to the square root of 2 is an example of a non-decreasing sequence that is bounded above and therefore converges.

Students do not need to know any of that as it will be developed in the lesson. Specifically, don’t even mention square roots, the square root of 2, or even irrational numbers until a student mention something of the sort.

This is not an efficient algorithm for finding square roots. There are far more efficient ways.

We begin with some preliminaries.

Preliminaries

  • There is a blank table that you can copy for students to use here.
  • There is a summary of the new terms used and completed table Sequence Notes. Do not give this out until after the lesson is completed.
  • We will be working with some rather long decimal numbers that will need to be squared. Scientific and graphing calculators usually compute with 14 digits and give their results rounded to 12 digits. Since ours will quickly get longer than that, I suggest you use WolframAlpha. This can be used with a computer online (at wolframalpha.com) or with an app available for smart phones and tablets. It is best if students have this website or app for their individual use.

Students will enter their numbers as shown below. Specifying “30 digits” will produces answers long enough for our purpose. To speed things up, students can edit the current number by changing the last digit in the entry line. When you get started you may have to show students how to do this. Students will need internet access.

  Computer                                                                                 Smart phone

The Lesson

The style of the lesson is Socratic. You, the teacher, will present the problem, explain how they are to go about it, and ask leading questions as appropriate. Some questions are suggested; be ready to ask others.  Later, you will have to explain (define) some new words, but as much as possible let the class suggest what to do. Drag things out of them, rather than telling them.

To begin – Produce some data

Explain to the class that they are going to generate and investigate two lists of numbers (technically called sequences). Each new member of the lists will be a number with one more decimal place than the preceding number.

The first list, whose members are called Ln, will be the largest number with the given number of decimal places, n, whose square is less than two. The subscript, n, stands for the number of decimal places in the number.

Ask: “What is the largest integer whose square is less than 2?” Answer 1, so, L0 = 1. Ask: What is the largest one place decimal whose square is less than 2?”  Answer L1 =1.4.

The second list, Gn, will be the smallest number whose square is greater than 2. So, G0 = 2 and G1 = 1.5. Notice that 1.42 = 1.96 < 2 and 1.52 = 2.25 > 2

Divide the class into 10 groups named Group 0, Group 1, Group 2, …, Group 9. In each round the groups will append their “name” to the preceding decimal and square the resulting number. Group 0 squares 1.40, group 1, squares 1.41, group 2 squares 1.42, etc. using WolframAlpha.

Ask which groups have squares less than two and enter the largest in Ln, the next number will be the smallest number whose square is greater than 2; enter it in Gn.

Complete the table by entering the largest number whose square is less than 2 in the Ln column and the smallest number whose square is greater than 2 in the Gn column. At each stage, each group appends their digit to the most recent Ln . Project or write the table on the board. Students may fill in their own copy. A completed table is here: Sequence Notes and definitions

Next – Lead a discussion

When the table is complete, prompt the students to examine the lists and come up with anything and everything they observe whether it seems important or not. Accept and discuss each observation and let the others say what they think about each observation. (Obviously, don’t deprecate or laugh at any answer – after all at this point, we don’t know what is and is not significant.)

There are (at least) three observations that are significant to what we will consider next. Hopefully, someone will mention them; keep questioning them until they do. They are these, although students may use other terms:

  1. Ln is non-decreasing. Students may first say Ln is increasing. Pause if they do and look at L12 and L13, and L15 and L16. Ask how they know Ln is non-decreasing (because each time we add a digit on the end, you get a bigger number).
  2. Likewise, Gn is non-increasing.
  3. For all n, Ln < Gn, and the numbers differ only in the last digit, and with the last digits differing by 1.

Direct instruction: Explain these ideas and terms (definition)

  • A sequence is a list or set of numbers in a given order.
  • A sequence is bounded above if there exists a number greater than or equal to all the terms of the sequence. The smallest upper bound of a sequence is called its least upper bound (l.u.b.)
  • A sequence is bounded below if there exists a number less than or equal to all the terms of the sequence. The largest lower bound is called the greatest lower bound (g.l.b.)

More questions: Apply these terms to the sequence Ln with questions like these:

  • Is Ln bounded above, below, or not bounded? (Bounded above)
  • Give an example of a number greater than all the terms of Ln. (Many answers: 1,000,000, 4, 2, 1.415, etc. and, in fact, any and every number in Gn)
  • What is the l.u.b. of Ln? Can you think of the smallest number that is an upper bound of this sequence? (Yes, \sqrt{2}. Don’t tell them this – drag it out of them if necessary.) Why? How do you know this?
  • Make the class convince you that for all n, \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{L}_{n}}} \right\}=\sqrt{2}

Ask similar questions about Gn.

  • Is Gn bounded above, below, or not bounded? (Bounded below)
  • Give an example of a number less than all the terms of Ln. (Many answers: any negative number, zero, 1, 1.414, etc. Any and every number in Ln)
  • What is the g.l.b. of Gn? Can you think of the greatest number that is a lower bound of this sequence? (Yes, \sqrt{2}) Why? How do you know this?
  • Make the class convince you that for all n, \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{G}_{n}}} \right\}=\sqrt{2}

 

Summing Up

Ask, “What’s happening with the numbers in the Ln sequence?” and “What’s happening to the numbers in the Gn sequence?”

The answer you want is that they are getting closer to \sqrt{2}, one from below, the other from above. (As always, wait for a student to suggest this and then let the others discuss it.)

Once everyone is convinced, explain how mathematicians say and write, “gets closer to”:

Mathematicians say that \sqrt{2} is the limiting value (or limit) of both sequences. They write \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{L}_{n}}} \right\}=\sqrt{2} and \displaystyle \underset{{n\to \infty }}{\mathop{{\lim }}}\,\left\{ {{{G}_{n}}} \right\}=\sqrt{2}.

Explain very carefully that while n\to \infty is read, “n approaches infinity,” that infinity, \infty , is not a number. The symbol n\to \infty means that n gets larger without bound or that n gets larger than all (any, every) positive numbers.

In a more technical sense there is an infinite series \displaystyle \sum\limits_{{n=0}}^{\infty }{{{{a}_{n}}\cdot {{{10}}^{{-n}}}}} where \displaystyle {{a}_{n}} is one of the digits 0, 1, 2, 3, …, 9, but there is no formula for listing the values of \displaystyle {{a}_{n}}. However, the sequence of partial sum of this series is the sequence \displaystyle \left\{ {{{L}_{n}}} \right\} which converges to \displaystyle \sqrt{2}. Therefore, \displaystyle \sum\limits_{{n=0}}^{\infty }{{{{a}_{n}}\cdot {{{10}}^{{-n}}}=\sqrt{2}}}

\displaystyle \sqrt{2} is an Irrational number, but this same procedure may be used to find decimal approximation of roots of rational numbers as well. However, for Rational numbers, there are easier ways.

Finally, Irrational numbers are exactly those that cannot be written as repeating (or terminating) decimals. They “go on forever” with no pattern. The decimals you can calculate eventually stop and are rounded to the last digit. Even WolframAlpha and similar computers must eventually do this. Irrational numbers are the limits of sequences like the one we looked at today.

Exercises

  1. Follow the procedure above to find the sequence whose limit is \sqrt{{\frac{{16}}{{121}}}} . Find this number the usual way (simplify and use long division) and compare the results.
  2. Follow the procedure above to find the sequence whose limit is \sqrt{{0.390625}} . Find this number the usual way and compare the results.
  3. Using WolframAlpha determine if the computer is using Ln, Gn. both, or neither when it gives a value for \sqrt{2}. (Hint: enter “square root 2 to 5 digits” and change to 6, 7, and 8 digits; compare the answer with the sequences, you found.)

Answers:

  1. 0.363636…
  2. 0.625
  3. For n = 5 and 6 the numbers are from Ln, for n = 7 and 8 they are from Gn. WolframAlpha is using a different algorithm to compute the square root of 2; the numbers appear from both sequences due to the rounding of the answers. To see WolframAlpha’s algorithm type “square root algorithm” on the entry line. This method also produces a sequence of approximations a/b.

Revised July 28, 2021