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Transforming Instruction in Undergraduate
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(TRIUMPHS)
Fall 2017
Gaussian Guesswork: In9nite Sequences and the Arithmetic-Gaussian Guesswork: In9nite Sequences and the Arithmetic-
Geometric Mean Geometric Mean
Janet Heine Barnett
Colorado State University-Pueblo
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Barnett, Janet Heine, "Gaussian Guesswork: In9nite Sequences and the Arithmetic-Geometric Mean"
(2017).
Calculus
. 2.
https://digitalcommons.ursinus.edu/triumphs_calculus/2
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Gaussian Guesswork:
Innite Sequences and the Arithmetic-Geometric Mean
Janet Heine Barnett
∗
November 10, 2021
Just prior to his 19th birthday, the mathematical genius Carl Friedrich Gauss (1777–1855) began a
“mathematical diary” in which he recorded his mathematical discoveries for nearly 20 years.
1
Among
these discoveries is the existence of a beautiful relationship between three particular numbers:
• the ratio of the circumference of a circle to its diameter, or π;
• a specic value of a certain (elliptic) integral, which Gauss denoted by ϖ = 2
1
0
dx
√
1 − x
4
;
• a number called “the arithmetic-geometric mean” of
√
2 and 1, which he denoted by M(
√
2, 1).
Like many of his discoveries, Gauss uncovered this particular relationship through a combination of
the use of analogy and the examination of computational data, a practice that historian Adrian Rice
calls “Gaussian Guesswork” in his Math Horizons
2
article subtitled “Why 1.19814023473559220744 . . .
is such a beautiful number” [Rice, November 2009].
This project is one of a set of three projects that looks at the power of Gaussian guesswork
via the story of his discovery of this beautiful relationship through excerpts from his mathematical
diary
3
and related manuscripts. This particular project focuses on how innite sequences are used
to dene the arithmetic-geometric mean. We begin in Section 1 with the denition, some examples
and basic properties of the arithmetic-geometric mean. In Section 2, we then look briey at how the
arithmetic-geometric mean is related to the Gaussian Guesswork story.
∗
Department of Mathematics and Physics, Colorado State University-Pueblo, Pueblo, CO 81001-4901;
1
Gauss’ extraordinary mathematical talent was recognized at a young age and nurtured by his teachers at St.
Katherine’s Public School in the family’s home region of Brunswick, Germany. The rst in his family to obtain a
higher education (his father worked at various trades and his mother was a housemaid before marriage), he completed
his degree in mathematics by age 18 and his doctorate by age 22. His academic studies and early research were made
possible by the patronage of Charles William Ferdinand, Duke of Brunswick (1735–1806), who paid Gauss’ educational
expenses and continued to provide him with a stipend until his own death in 1806. Gauss acknowledged the generosity
of the Duke—whom he addressed as “Most Serene Prince”—in the dedication of his famous groundbreaking book,
Disquisitiones Arithmetica (Number Theory Investigations), published in 1801. In recognition of his own generous
contributions to the eld, Gauss has become known as the “Prince of Mathematics.” During his lifetime, he made
important discoveries in every area of mathematics that was then known, as well as a few new ones that he helped
to create, including elliptic functions. He also applied his mathematical insights to problems in physics, geodesy,
magnetism, optics and astronomy. For more about Gauss’ life and works, see the references [Dunnington, 2004],
[O’Connor and Robertson, 1996] and [Famous Scientists, 2016].
2
Math Horizons is the undergraduate magazine of the Mathematical Association of America (MAA). It publishes
expository articles about mathematics and the culture of mathematics, including mathematical people, institutions, hu-
mor, games and book reviews. For more information, visit https://www.maa.org/press/periodicals/math-horizons.
3
Gauss’ diary remained in the possession of his family until 1898 and was rst published by Felix Klein (1849–1925)
in [Klein, 1903]. An English translation with commentary on its mathematical contents by historian of mathematics
Jeremy Gray appears in [Gray, 1984], and was later reprinted in [Dunnington, 2004, 469–496]. A facsimile of the
original diary can also be found in Volume X.1 of Gauss’ Werke (Collected Works).
1
1 The Arithmetic-Geometric Mean
Although Gauss appears to have discovered the arithmetic-geometric mean when he was only 14
years old,
4
he published very little about it during his lifetime. Much of what we know about his
work in this area instead comes from a single paper [Gauss, 1799] that was published as part of his
mathematical legacy (or, as the Germans would say, as part of his Nachlass)only after his death.
In the rst excerpt
5
from this paper that we will read in this project, Gauss began by dening two
related innite sequences.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Let
a, a
1
, a
2
, a
3
, . . .
b, b
1
, b
2
, b
3
, . . .
be two sequences
6
of magnitudes formed by this condition:
that the terms of either correspond to the mean between the preceding terms, and indeed,
the terms of the upper sequence have the value of the arithmetic mean, and those of the
lower sequence, the geometric mean, for example,
a
1
=
1
2
(a + b), b
1
=
√
ab, a
2
=
1
2
(a
1
+ b
1
), b
2
=
a
1
b
1
, a
3
=
1
2
(a
2
+ b
2
), b
3
=
a
2
b
2
.
But we suppose a and b to be positive reals and [that] the quadratic [square] roots are
everywhere taken to be the positive values; by this agreement, the sequences can be produced
so long as desired, and all of their terms will be fully determined and positive reals [will be]
obtained.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 1 Write (at least) two comments and (at least) two questions about Gauss’ description
of the arithmetic-geometric mean in this excerpt.
Since this may be the rst time that you are hearing about the ‘geometric mean’ of two positive
numbers a, b, notice that one way to think about it is simply as the length s of a side of a square that
has the same area as a rectangle with sides of length a and b respectively; this gives us the formula
s
2
= ab, or s =
√
ab. Along with the more familiar ‘arithmetic mean,’ computed via the formula
1
2
(a + b), the geometric mean is one of several ‘averages’ that were studied by mathematicians as
far back as the ancient Greeks. The terminology ‘arithmetic’ and ‘geometric’ to describe these two
dierent types of means also dates back to ancient Greek mathematics.
In his 1799 Nachlass paper, Gauss gave four specic examples of sequences (a
n
) and (b
n
) dened
by way of the arithmetic and geometric means as described above; let’s pause to look at the rst two
of these now.
4
Gauss himself reminisced about his 1791 discovery of this idea in a letter, [Gauss, 1816], that he wrote to his friend
Schumacher much later. Although his memory of the exact date of his discovery may not have been accurate when
he wrote that 1816 letter, Gauss was certainly familiar with the arithmetic-geometric mean by the time he began his
mathematical diary in 1796.
5
All translations from [Gauss, 1816] used in this project were prepared by George W. Heine III (Math and Maps),
2017.
6
Gauss himself used prime notation (i.e., a
′
, a
′′
, a
′′′
) to denote the terms of the sequence. In this project, we instead
use indexed notation (i.e., a
1
, a
2
, a
3
) in keeping with current notational conventions. To fully adapt Gauss’ notation
to that used today, we could also write a
0
= a and b
0
= b.
2
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Example 1: a = 1, b = 0.2
a = 1.00000 00000 00000 00000 0 b = 0.20000 00000 00000 00000 0
a
1
= 0.60000 00000 00000 00000 0 b
1
= 0.44721 35954 99957 93928 2
a
2
= 0.52360 67977 49978 99964 1 b
2
= 0.51800 40128 22268 36005 0
a
3
= 0.52080 54052 86123 66484 5 b
3
= 0.52079 78709 39876 24344 0
a
4
= 0.52080 16831 12999 95414 3 b
4
= 0.52080 16380 99375
a
5
= 0.52080 16381 06187 b
5
= 0.52080 16381 06187
Here a
5
, b
5
dier in the 23
rd
decimal place.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 2 This task examines Example 1 from Gauss’ paper.
(a) Verify that the values given by Gauss in the previous excerpt are correct. Are
you able to use your calculator to obtain the same degree of accuracy (21 decimal
places!) that Gauss obtained by hand calculations?
(b) Write three observations about the two sequences in this example.
Use a full sentence to state each of your observations.
Task 3 In Example 2 from his paper, Gauss set a = 1, b = 0.6.
(a) Use a calculator or computer to compute the next four terms of each sequence.
a = 1.00000 00000 00000 00000 0 b = 0.60000 00000 00000 00000 0
a
1
= b
1
=
a
2
= b
2
=
a
3
= b
3
=
a
4
= b
4
=
(b) In Task 2, part (b), you made three observations about Gauss’ Example 1.
Do the same general patterns hold for Example 2?
If so, why do you think this is? If not, in what way(s) are the two examples
dierent?
With these two examples in hand, let’s now go back to look at some of the general properties
that Gauss claimed will always hold for such sequences. The following excerpt includes three such
properties, which we will examine in further detail in the tasks below. For now, read through each
of these carefully and compare them to the observations that you made in Task 2, part (b).
3
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Moreover, we rst oer here the following observations:
I. If a = b, all of the terms of either sequence will be = a = b.
II. If however a, b are unequal, then (a
1
−b
1
)(a
1
+ b
1
) =
1
4
(a −b)
2
, whence it is concluded
that b
1
< a
1
, and also that b
2
< a
2
, b
3
< a
3
etc., i.e. any term of the lower sequence
will be smaller than the corresponding [term] of the upper. Wherefore, in this case, we
suppose also that b < a.
III. By the same supposition it will be that a
1
< a, b
1
> b, a
2
< a
1
, b
2
> b
1
etc.; therefore
the upper sequence constantly decreases, and the lower constantly increases; thus it is
evident that each [sequence] has a limit; these limits are conveniently expressed
7
a
∞
, b
∞
.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
The rst of Gauss’ observations needs little proof—if a = b, then clearly the arithmetic mean is
1
2
(a + b) =
1
2
(2a) = a and the geometric mean is
√
ab =
√
a
2
= a; thus, (a
n
) = (a) = (b
n
) gives just
one constant sequence. In the following tasks, you will verify Gauss’ next two observations.
Task 4 This task examines Gauss’ Observation II:
II. If however a, b are unequal, then (a
1
− b
1
)(a
1
+ b
1
) =
1
4
(a − b)
2
, whence
it is concluded that b
1
< a
1
, and also that b
2
< a
2
, b
3
< a
3
etc., i.e. any
term of the lower sequence will be smaller than the corresponding [term]
of the upper. Wherefore, in this case, we suppose also that b < a.
(a) Use the denitions of a
1
, b
1
to verify that (a
1
− b
1
)(a
1
+ b
1
) =
1
4
(a − b)
2
.
(b) Now explain why the equality in part (a) allows us to conclude that a
1
> b
1
.
(c) Finally, explain why we are now able to conclude that b
n
< a
n
for all values of n.
7
In keeping with the notation currently in use today, we have replaced Gauss’ use of superscripts (a
∞
, b
∞
) to denote
the limiting values with subscripts (a
∞
, b
∞
) throughout this project.
4
Task 5 This task examines Gauss’ Observation III:
III. By the same supposition it will be that a
1
< a, b
1
> b, a
2
< a
1
,
b
2
> b
1
etc.; therefore the upper sequence constantly decreases, and the
lower constantly increases; thus it is evident that each [sequence] has a
limit; these limits are conveniently expressed by a
∞
, b
∞
.
To see what Gauss meant by the phrase “the same supposition” here, look back at his
statement of Observation II, and notice that he ended that earlier observation with
the assertion that “. . . we suppose also that b < a.”
(a) Use the assumption that b < a and the denitions of the two sequences to verify
Gauss’ claim that (a
n
) is a strictly decreasing sequence and that (b
n
) is a strictly
increasing sequence.
Consider Gauss’ next assertion: “thus it is evident that each [sequence] has a
limit.” Write a convincing explanation why this conclusion must hold, given
what we know about these two sequences thus far. Do you agree with Gauss that
this conclusion is ‘evident’?
(b) In your answer to part (a), you may have mentioned that the two sequences in
question are bounded—that is, bounded both above and below. Whether or not
you did so, state the values of the upper and lower bounds for the sequence (a
n
),
and for the sequence ( b
n
). Why is it important that (a
n
) is bounded, as well
as decreasing? Similarly, why is it important that (b
n
) is bounded, as well as
increasing?
(c) Now nd a theorem in your Calculus textbook (in the chapter that considers
innite sequences) that could also be used to conclude that the sequences (a
n
)
and (b
n
) converge. Give both the name of this theorem and its full statement.
(d) Do you think that the notation a
∞
, b
∞
that Gauss used to denote the limits of
these sequences is appropriate? Why or why not? Based on what you’ve seen in
Examples 1 and 2, how do you expect the values of a
∞
and b
∞
to be related?
Gauss made one nal observation about the sequences (a
n
), (b
n
), which you have perhaps already
predicted yourself:
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
IV. Finally, from
a
1
−b
1
a−b
=
(a−b)
4(a
1
+b
1
)
=
a−b
2(a+b)+4b
1
, it follows that a
1
− b
1
<
1
2
(a − b), and in
the same way, a
2
− b
2
<
1
2
(a
1
− b
1
), etc. Hence, it is concluded that a − b, a
1
− b
1
,
a
2
− b
2
, a
3
− b
3
etc. forms a strictly decreasing sequence and the limit itself is = 0.
Thus a
∞
= b
∞
, i.e., the upper and lower sequences have the same limit, which always
remains below the one and above the other.
We call this limit the arithmetic-geometric mean between a and b, and denote it by
M(a, b).
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
5
Task 6 This task examines Gauss’ Observation IV from the previous excerpt.
(a) Verify that
a
1
−b
1
a−b
=
(a−b)
4(a
1
+b
1
)
. Hint: Start with the equality from Gauss’
Observation II: (a
1
−b
1
)(a
1
+b
1
) =
1
4
(a−b)
2
.
(b) Now verify that
(a−b)
4(a
1
+b
1
)
=
a−b
2(a+b)+4b
1
.
(c) Combining the results of parts (a) and (b), we now have
a
1
−b
1
a−b
=
a−b
2(a+b)+4b
1
.
Use this equality to explain why the following hold:
(i) a
1
− b
1
<
1
2
(
a
−
b
)
(ii) a
2
− b
2
<
1
2
(a
1
− b
1
) <
1
4
(a − b)
(iii) Taking n to be any arbitrary value, a
n
− b
n
<
1
2
n
(a − b)
(d) Based on part (c), why can we now conclude (with Gauss) that ’a − b, a
1
− b
1
,
a
2
− b
2
, a
3
− b
3
etc forms a strictly decreasing sequence and the limit itself is = 0’?
That is, why does lim
n→∞
(a
n
− b
n
) = 0? Also explain why this allowed Gauss to
conclude that a
∞
= b
∞
.
Now that we have veried Gauss’ claim that a
∞
= b
∞
, his denition of the arithmetic-geometric
mean M(a, b) of two positive numbers a and b as the common value of those two limits makes sense!
Stating this in modern notation, we have:
M(a, b) = lim
n→∞
a
n
= lim
n→∞
b
n
, where a
0
= a, b
0
= b, a
n+1
=
1
2
(a
n
+ b
n
) and b
n+1
=
√
a
n
b
n
.
This is precisely how M(a, b) is still dened today, although it is often denoted instead by agm(a, b) or
agM(a, b). Today, the arithmetic-geometric mean is used to construct fast algorithms for calculating
values of elementary transcendental functions and some classical constants, like π — examples 1 and
2 above show just how rapidly this process generally converges. In the next section of this project,
we will consider why Gauss himself became interested in the arithmetic-geometric mean.
2 Why is 1. 19814023473559220744 . . . such a beautiful number?
The complete answer to this question requires a bit more mathematics than we will study in this
project. In this closing section, we summarize just the highlights. We begin by looking at Gauss’
fourth example of the arithmetic-geometric mean in his 1799 Nachlass paper.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Example 4: a =
√
2, b = 1
a = 1.41421 35623 73095 04880 2 b = 1.00000 00000 00000 00000 0
a
1
= 1.20710 67811 86547 52440 1 b
1
= 1.18920 71150 02721 06671 7
a
2
= 1.19815 69480 94634 29555 9 b
2
= 1.19812 35214 93120 12260 7
a
3
= 1.19814 02347 93877 20908 3 b
3
= 1.19814 02346 77307 20579 8
a
4
= 1.19814 02347 35592 20744 1 b
4
= 1.19814 02347 35592 20743 9
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
6
Notice that the rst 20 decimal places of the limit value for this example are precisely the number
that appears in title of this section. But why would Gauss or others have considered M(
√
2, 1) to
be particularly beautiful?
In fact, Gauss’ initial acquaintance with the number 1.19814023473559220744 . . . had nothing to
do with the arithmetic-geometric mean, but was related instead to his eorts to evaluate certain
types of integrals. One such integral and its value will be quite familiar to you; namely,
1
0
1
√
1 − t
2
dt =
π
2
Make sure you believe that this is a correct value . . . and then consider what you might do to evaluate
the following quite-similar-looking integral:
1
0
1
√
1 − t
4
dt
Stuck? If so, then you’re in good company! Evaluating integrals of the form
x
0
1
√
1−t
n
dt for n > 2
is notoriously dicult
8
. . . but also necessary for problems such as determining the arc length of
ellipses and other curves that naturally arise in astronomy and physics. For instance, the integral
1
0
1
√
1−t
4
dt gives the arc length of a curve known as a lemniscate.
9
Based on entries in his mathematical diary, we know that Gauss himself began to study elliptic
integrals in September 1796, at the ripe old age of 19. Following an analogy suggested by the fact
that π = 2
1
0
1
√
1−t
2
dt, he dened a new constant by setting ϖ = 2
1
0
1
√
1−t
4
dt and used power series
techniques to calculate ϖ to twenty decimal places, nding that
10
ϖ = 2.622057055429211981046 . . . .
Task 7 This task looks at the relationship between ϖ, π, and M(
√
2, 1).
Recalling that
π = 3.14159265258979323846 . . .
and
M(
√
2, 1) = 1.19814023473559220744 . . . ,
try to nd a relationship between these two numbers and the value of ϖ given above.
Three years after taking up the study of elliptic functions, Gauss wrote the following entry in his
mathematical diary on May 30, 1799:
8
For n = 3 and n = 4,
∫
1
0
1
√
1−t
n
dt is called an elliptic integral; for n ≥ 5, it is called hyperelliptic.
9
Additional detail about how Gauss connected the arithmetic-geometric mean to the evaluation of the integral
∫
1
0
1
√
1−t
4
dt and about the connection of that integral to the lemniscate can be found in the author’s projects Gaus-
sian Guesswork: Elliptic Integrals and Integration by Substitution and Gaussian Guesswork: Polar Coordinates, Arc
Length and the Lemniscate Curve, respectively; both projects are available at https://digitalcommons.ursinus.edu/
triumphs_calculus.
10
The symbol ‘ϖ’ that Gauss used to denote this specic value is called “varpi”; it is a variant of the Greek letter π.
7
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
We have established that the arithmetic-geometric mean between
√
2 and 1 is π/ϖ to 11
places; the proof of this fact will certainly open up a new eld of analysis.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
It took Gauss another year to fully prove that his guesswork about the numerical relationship
M(
√
2, 1) =
π
ϖ
was correct. The “new eld of analysis” that opened up in connection with this
proof led him well beyond the study of elliptic functions of a single real-valued variable, and into
the realm of functions of several complex-valued variables. Today, a special class of such functions,
known as the ‘theta functions,’ provides a powerful tool that is used in a wide range of applications
throughout mathematics—providing yet one more piece of evidence of Gauss’ extraordinary ability
as a mathematician and a guesswork genius!
References
David Cox. The Arithmetic-Geometric Mean of Gauss. L’Enseignement Mathématique, 30:275–330,
1984.
Guy Waldo Dunnington. Carl Friedrich Gauss: Titan Of Science. The Mathematical Association
of America, Washington DC, 2004. Reprint of original 1955 publication. Includes the English
translation of Gauss’ Diary by Jeremy Gray.
Famous Scientists. Carl Friedrich Gauss. 2016. www.famousscientists.org/
carl-friedrich-gauss/. Accessed: November 7, 2021.
Carl Friedrich Gauss. Arithmetisch Geometrisches Mittel (Arithmetic-Geometric Mean). 1799. In
Ernst Schering, editor, Werke, volume III, pages 361–432. Gedruckt in der Dieterichschen univer-
sitätsdruckerei, Göttingen, 1866.
Carl Friedrich Gauss. Letter from Gauss to Schumacher dated April 1816 (in German). 1816. In
Felix Klein, editor, Werke, volume X.1, pages 247–248. Konigliche Gesellschaft der Wissenschaft,
Göttingen, 1917.
J. J. Gray. A Commentary on Gauss’s Mathematical Diary, 1796–1814, with an English Translation.
Expositiones Mathematicae, 2(2):97–130, 1984.
Felix Klein. Gauss’ Wissenschaftliches Tagebuch (Mathematical Diary), 1796–1814. Mathematische
Annalen, 57:1–34, 1903.
J. J. O’Connor and E. F. Robertson. Johann Carl Friedrich Gauss. MacTutor, 1996. https:
//mathshistory.st-andrews.ac.uk/Biographies/Gauss/. Accessed: November 7, 2021.
Adrian Rice. Gaussian Guesswork, or why 1.19814023473559220744 . . . is such a beautiful number.
Math Horizons, pages 12–15, November 2009.
8
Notes to Instructors
PSP Content: Topics and Goals
This Primary Source Project (PSP) is one of a set of three 1–2 day projects designed to consolidate
student prociency of the following traditional topics from a rst-year Calculus course:
• Gaussian Guesswork: Elliptic Integrals and Integration by Substitution
• Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve
• Gaussian Guesswork: Innite Sequences and the Arithmetic-Geometric Mean
Each of these PSPs can be used either alone or in conjunction with any of the other two. All three
are based on excerpts from Gauss’ mathematical diary and related primary texts that will introduce
students to the power of numerical experimentation via the story of his discovery of a relationship
between three particular numbers: the ratio of the circumference of a circle to its diameter (π);
a specic value (ϖ) of the elliptic integral u =
x
0
dt
√
1 − t
4
; and the Arithmetic-Geometric Mean
of
√
2 and 1. Like many of his discoveries, Gauss uncovered this particular relationship through a
combination of the use of analogy and the examination of computational data, a practice referred
to as “Gaussian Guesswork” by historian Adrian Rice in his Math Horizons article subtitled “Why
1.19814023473559220744 . . . is such a beautiful number” [Rice, November 2009].
The primary content goal of this particular PSP is to consolidate students’ understanding of
sequence convergence, and especially the Monotone Convergence Theorem.
Student Prerequisites
In light of the project’s content goals, it is assumed that students have had some introduction to the
study of sequence convergence, including the statement of the Monotone Convergence Theorem, and
that they are familiar with the associated notation and basic vocabulary. Familiarity with integration
techniques sucient to conrm that
1
0
1
√
1−t
2
dt =
π
2
will also be useful for reading the closing section
of the project; however, evaluation of this or any other integral is not required for completion of the
student tasks in this particular PSP.
PSP Design and Task Commentary
This project begins with a brief introduction to the historical context of Gauss’ discovery. In Section
1, students then read Gauss’ denition of the two sequences needed for a denition of the arithmetic-
geometric mean, and work through two of his numerical examples for such sequences. They then
read Gauss’ observations about these examples, and complete proofs of the general properties that
he derived from those observations. In Task 5 (based on Gauss’ Observation III), the Monotone
Convergence Theorem is invoked in order to conclude that each of the two sequences in question
does converge. In Task 6 (based on Gauss’ Obervation IV), students then prove that these limits are
the same, a fact that naturally leads to the denition of the arithmetic-geometric mean. By way of
conclusion, Section 2 then ties the mathematical details of this particular project to the larger story
of Gauss’ amazing discovery.
Suggestions for Classroom Implementation
Classroom implementation of this and other PSPs in the collection may be accomplished through
individually assigned work, small-group work and/or whole-class discussion; a combination of these
instructional strategies is recommended in order to take advantage of the variety of questions included
in the project. To reap the full mathematical benets oered by the PSP approach, students should
9
be required to read assigned sections in advance of in-class work, and to work through primary source
excerpts together in small groups in class.
The author’s method of ensuring that advance reading takes place is to require student completion
of “Reading Guides” (or “Entrance Tickets”), for which students receive credit for completion, but
with no penalty for errors in solutions. A typical Reading Guide will include “Classroom Preparation”
exercises (drawn from the PSP Tasks) for students to complete prior to arriving in class. They may
also include “Discussion Questions” that ask students only to read a given task and jot down some
notes in preparation for class work. On occasion, other tasks are assigned as follow-up to a prior
class discussion.
In completing assigned Reading Guides, students are asked to strive to answer each question
correctly, but to think of the guide as preparatory work for class, not as a nal product (e.g., formal
polished write-ups are not expected). Students who arrive unprepared to discuss assignments on
days when group work is conducted based on advance reading are not allowed to participate in those
groups, but are allowed to complete the in-class work independently. Reading Guides are collected at
the end of each class period for instructor review and scoring (again, for completeness only), thereby
providing helpful feedback to the instructor about individual and whole-class understanding of the
material, in addition to supporting students’ advance preparation eorts.
A sample Reading Guide (based on the Day 1 Advanced Preparation Work suggested in the
Sample Implementation Schedule given below) appears in the Appendix at the end of these Notes.
L
A
T
E
X code of the entire PSP is available from the author by request to facilitate preparation of
reading guides or ‘in-class task sheets’ based on tasks included in the project. The PSP itself can
also be modied by instructors as desired to better suit their goals for the course.
Sample Implementation Schedule (based on a 75-minute class period)
To complete this project in its entirety, the following implementation schedule is recommended.
Instructors teaching 50-minute class periods can easily adapt this to a 2-day schedule as well, with
small-group work on Section 1 tasks continuing into Day 2.
• Advance Preparation Work (to be completed before class):
Read pages 1–3, completing Tasks 1–3 for class discussion along the way, per the sample
Reading Guide in the Appendix to these Notes.
• Day 1: In-Class Work
– Brief whole-class or small-group comparison of answers to Tasks 1–3.
– Small-group work on Tasks 4–6 (supplemented by whole-class discussion as deemed ap-
propriate by the instructor). Note that these three tasks form the mathematical core of
this PSP. They are based on the excerpt at the top of page 4, which is recommended
for advance preparation work. If the advance preparation assignment does not include
reading the top of page 4, then students will need additional class time to read that half
page prior to starting on Task 4.
– Time permitting, individual or small-group reading of last half of page 6 (below Task 6).
• Follow-up Assignment (to be completed prior to the next class period):
As needed, read the last half of page 6; also read Section 2 (pages 6–8), completing Task 7
along the way. This assignment could also be made part of an advance preparation “Reading
Guide” if a second day of in-class small-group work is desired. Note, however, that the answer
to Task 7 is revealed at the top of page 8.
10
• Day 2 (optional): Closing Discussion (5–30 minutes)
– In addition to a brief (5–10 minute) whole-class discussion of Section 2 and/or student
questions on Section 1, students could be asked to share how they approached Task 7.
(Again, the answer to that Task is revealed at the top of page 8, but how students
approached it could make for interesting discussion.)
– Instructors who wish to share additional details about the story told in this project during
their closing comments can nd those in the author’s other two “Gaussian Guesswork”
PSPs; adding this further commentary will, of course, take up more class time.
• Homework (optional): A complete formal write-up of student work on Tasks 3–5 could be
assigned, to be due at a later date (e.g., one week after completion of the in-class work).
Connections to other Primary Source Projects
Links to all three “Gaussian Guesswork’’ PSPs (described earlier in these Notes) are as follows:
• Gaussian Guesswork: Elliptic Integrals and Integration by Substitution
https://digitalcommons.ursinus.edu/triumphs_calculus/8/
• Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve
https://digitalcommons.ursinus.edu/triumphs_calculus/3/
• Gaussian Guesswork: Innite Sequences and the Arithmetic-Geometric Mean
https://digitalcommons.ursinus.edu/triumphs_calculus/2/
The following additional projects based on primary sources are also freely available for use in
teaching standard topics in the calculus sequence. The PSP author name of each is given (together
with the general content focus, if this is not explicitly given in the project title). Each of these can be
completed in 1–2 class days, with the exception of the four projects followed by an asterisk (*) which
require 3, 4, 3, and 6 days respectively for full implementation. Classroom-ready versions of these
projects can be downloaded from https://digitalcommons.ursinus.edu/triumphs_calculus.
• Investigations Into d’Alembert’s Denition of Limit (calculus version), Dave Ruch
• L’Hôpital’s Rule, Daniel E. Otero
• The Derivatives of the Sine and Cosine Functions, Dominic Klyve
• Fermat’s Method for Finding Maxima and Minima, Kenneth M Monks
• Beyond Riemann Sums: Fermat’s Method of Integration, Dominic Klyve
• How to Calculate π: Buon’s Needle (calculus version), Dominic Klyve (integration by parts)
• How to Calculate π: Machin’s Inverse Tangents, Dominic Klyve (innite series)
• Euler’s Calculation of the Sum of the Reciprocals of Squares, Kenneth M Monks (innite series)
• Fourier’s Proof of the Irrationality of e, Kenneth M Monks (innite series)
• Jakob Bernoulli Finds Exact Sums of Innite Series (Calculus Version),* Daniel E. Otero and
James A. Sellars
• Bhāskara’s Approximation to and Mādhava’s Series for Sine, Kenneth M Monks (approxima-
tion, power series)
• Braess’ Paradox in City Planning: An Application of Multivariable Optimization, Kenneth
M Monks
• Stained Glass, Windmills and the Edge of the Universe: An Exploration of Green’s Theorem,*
Abe Edwards
11
• The Fermat-Torricelli Point and Cauchy’s Method of Gradient Descent,* Kenneth M Monks
(partial derivatives, multivariable optimization, gradients of surfaces)
• The Radius of Curvature According to Christiaan Huygens,* Jerry Lodder
Acknowledgments
The development of this project has been partially supported by the Transforming Instruction in
Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Project with funding
from the National Science Foundation’s Improving Undergraduate STEM Education Program under
Grant Number 1523494. Any opinions, ndings, conclusions or recommendations expressed in this
project are those of the author and do not necessarily represent the views of the National Science
Foundation.
With the exception of excerpts taken from published translations of
the primary sources used in this project and any direct quotes from
published secondary sources, this work is licensed under a Creative
Commons Attribution-ShareAlike 4.0 International License (https:
//creativecommons.org/licenses/by-sa/4.0/legalcode). It al-
lows re-distribution and re-use of a licensed work on the conditions
that the creator is appropriately credited and that any derivative work
is made available under “the same, similar or a compatible license.”
For more information about TRIUMPHS, visit:
https://blogs.ursinus.edu/triumphs/
12
APPENDIX
This appendix provides a ‘Sample Reading Guide’ that illustrates the author’s method
for assigning advance preparation work in connection with classroom implementation of
primary source projects. More detail concerning these guides is included in the subsection
“Suggestions for Classroom Implementation” of the Notes to Instructors for this project.
Note that the full text of each assigned Task is reproduced on the guide, with blank
space for students’ responses deliberately left below each question. This not only makes
it easier for students to jot down their thoughts as they read, but also makes their notes
more readily available to them during in-class discussions. This practice also makes it
easier for the instructor to eciently review each guide for completeness, and for students
to review their own notes and instructor feedback once it is returned to them.
SAMPLE READING GUIDE: Advance Preparation Work for Day 1
Background Information: The goal of the advance reading and tasks assigned in this 3-page reading guide
is to familiarize students with the denition and examples of the two sequences that dene the arithmetic-
geometric mean in order to prepare them for in-class small group work on Tasks 4–6.
**********************************************************************************************
Reading Assignment: Gaussian Guesswork: Sequences and the Arithmetic-Geometric Mean, pp. 1–3
1. Read the introduction on page 1.
Any questions or comments?
2. Begin reading Section 1, stopping att Task 1.
Any questions or comments?
3. Class Prep: Complete Task 1 here:
Task 1 Write (at least) two comments and (at least) two questions about Gauss’ description of
the arithmetic-geometric mean in the excerpt on page 2.
Sample Reading Guide for Day 1: Gaussian Guesswork: Innite Sequences and the Arithmetic-Geometric Mean 1
4. Continue your reading of Section 1, stopping at Task 2.
Any questions or comments?
5. Class Prep: Complete Task 2 here:
Task 2 This task examines Example 1 from Gauss’ paper.
(a) Verify that the values given by Gauss in the previous excerpt are correct. Are you able to
use your calculator to obtain the same degree of accuracy (21 decimal places!)
that Gauss obtained by hand calculations?
(b) Write three observations about the two sequences in this example.
Use a full sentence to state each of your observations.
Sample Reading Guide for Day 1: Gaussian Guesswork: Innite Sequences and the Arithmetic-Geometric Mean 2
6. Class Prep: Complete Task 3 here:
Task 3 In Example 2 from his paper, Gauss set a = 1 , b = 0.6.
(a) Use a calculator or computer to compute the next four terms of each sequence.
a = 1.00000 00000 00000 00000 0 b = 0.60000 00000 00000 00000 0
a
1
= b
1
=
a
2
= b
2
=
a
3
= b
3
=
a
4
= b
4
=
(b) In Task 1, part (b), you made three observations about Gauss’ Example 1.
Do the same general patterns hold for Example 2?
If so, why do you think this is? If not, in what way(s) are the two examples dierent?
7. Read the rest of page 3, starting below Task 3; also read the Gauss excerpt at the top of page 4.
Write at least one question or comment about this part of the reading.
Sample Reading Guide for Day 1: Gaussian Guesswork: Innite Sequences and the Arithmetic-Geometric Mean 3
