Math IA - Evolution of Sequences and Series 1 003159-0025
March 23, 2015
IB Mathematics HL
Internal Assessment
“The Evolution of Sequences and Series: A
study between eras with respect to Infinite
Sequences and Series and its various
applications in Math.”
Name: Kethan Reddy
Candidate Number: 003159 - 0025
School: American International School Chennai
Date: May, 2015
Session: May 2015
Math IA - Evolution of Sequences and Series 2 003159-0025
March 23, 2015
Contents Ancient Greek: 323 – 212 BC ......................................................................................................... 4
Euclid’s Definition of Prime Numbers........................................................................................ 4
Utilizing the Cultural Lens .......................................................................................................... 5
Archimedes’s Infinite Parabolic Summation .............................................................................. 5
Resurgence of Mathematics in the West (Modern Era) .................................................................. 8
Advancements in Algebra and the fall of Islam .......................................................................... 8
Convergence, Divergence and Riemann-Zeta Function ............................................................. 9
Natural Numbers Summation Proof .......................................................................................... 11
Romanticism in Mathematics and Science................................................................................ 13
Prime Product Formula Derivation ........................................................................................... 14
In Conclusion… ............................................................................................................................ 15
Bibliography ................................................................................................................................. 16
Appendix ....................................................................................................................................... 17
Convergent Table and Graph .................................................................................................... 17
Divergent Table and Graph ....................................................................................................... 17
Math IA - Evolution of Sequences and Series 3 003159-0025
March 23, 2015
Sequences and series have always puzzled me, regardless of being deceitfully simplistic or
bafflingly complex. Mathematics has, at its core, been the study of patterns in logic and
fundamental reasoning - Sequences and Series pushes this notion of mathematics to its very
edge, taking pattern recognition one step further to infinity. In this study, I will detail the
evolution of understanding behind Sequences and Series starting from the Ancient Greeks in 300
B.C to our modern understanding of this area of math in the 21st century, and how the
progression of this growth in understanding was sparked by individuals who were heavily
influenced by the culture of the time from the likes of Gregory, Taylor, Euler and Gauss. Before
we continue with our ethno-cultural study of the evolution of Sequences and Series, we must first
define these terms.
A sequence, for all intents and purposes, is a string of integers. A series is the sum of the
terms of a sequence in its crudest definition. Each term in a specific series are often produced by
a certain function, this may be in the form of formula or an algorithm. If this string of summation
is finite, then it is rightfully called a finite series. If there are an infinite number of terms, then it
is referred to as an infinite series. Finite summations are easier to handle than that of its
counterpart (infinite summations), because infinite series requires the use of mathematical
analysis and the notions of analytical continuation to be explored and understood fully. However,
we had to naturally progress to from finite sequences and series to infinite ones, and to fully
grasp the latter we must understand the origin of finite sequences and series, which could be
definitively traced back to an Ancient Greek named Euclid in his book called ‘Elements’.
‘Elements’ being a monumental book, laying the groundwork for mathematics as a whole.
Math IA - Evolution of Sequences and Series 4 003159-0025
March 23, 2015
Ancient Greek: 323 – 212 BC
Euclid’s Definition of Prime Numbers Euclid, or ‘Euclid of Alexandria’, was a Greek mathematician who is often referred to as
the father of geometry. He made significant contributions in spherical geometry, conic sections
and number theory. In number theory, we learn that prime numbers are the builing blocks of all
other numbers because of the fundamental theorem of arithmetic - every integer greater than 1
either is prime itself or is the unique product of prime numbers. In his magnus opus, Elements, he
provides the first instance of defining the prime number1 in definition 11 - A prime number is
that which is measured by a unit alone. How is this related to sequences and series? Well, Euclid
essentially defined the parameters of a prime number to a summation of units (or 1s). He used
visual representations to carry out this idea –
Let us take this as one ‘unit’ block.
Now let us take a prime number, 5, written as a summation of 5 unit blocks.
According to Euclid, no other number can sum itself to reach 5. For example, let us try
the case of summing 2. Let us try to sum 2, three times. (In other words, 2 × 3)
1 There has been evidence that the Egyptians knew about the prime numbers before hand, but this is the first definitive instance.
Math IA - Evolution of Sequences and Series 5 003159-0025
March 23, 2015
We see that it 2 cannot be added three times to fit into five, and we soon find out that it
will not work with 3 and four also. So, the only way to measure a prime number is by a unit
alone (or the prime number itself).
Utilizing the Cultural Lens This is the most fundamental summation theory in its time, using a function of
summation to define the attributes of other numbers. This visual representation of summation and
proofs is one that was unique to the Greek culture at the time – many worshipping the value of
math for its ‘real life application and connection to the physical world’. This time period could
be the birthplace, the spark, of the modern day sciences via various revolutions such as scientific
revolutions in the Islamic States, the Renaissance and the Enlightenment. The philosophy at the
time was one that was dominated by reason and inquiry, and scholars at the time wanted physical
and objective evidence. All theories, whether it be scientific or mathematic, had to be verified
with one of our senses – mostly the tactile and visual senses. This use of sensory perception to
validate mathematics was a calling-card of most Ancient Greek philosophers and thinkers. In a
way, they were uncovering truths about the structures in the known observable Universe, finding
patterns in shapes and geometries that held true because of the very nature of the shapes
themselves, for example the nature of prime numbers with respect to lines and the area of a given
parabolic sector. This is why Euclid and other Greek philosophers relied so heavily on visual aid
to provide mathematical evidence for their proofs, it was ingrained into the philosophy of the
culture. Euclid was not the only prominent Greek mathematician of the era, there were more
Greek thinkers that were influential – such as Archimedes.
Archimedes’s Infinite Parabolic Summation Archimedes was once the most influential thinkers at the time in the field of mathematics.
Math IA - Evolution of Sequences and Series 6 003159-0025
March 23, 2015
One of the first recorded incidences of the use of an infinite series is written by Archimedes in
the 3rd century BC, in a letter called ‘The Quadrature of the Parabola’. In this letter, he explained
that the area of a parabolic segment is 4
3 of an inscribed triangle (in this case the dark blue
triangle in fig. 2), in which he uses a geometric proof.
The following summation of an infinite series to prove the area of a parabolic segment is
4
3 an inscribed triangle goes as follows –
The green triangle in fig. 2 is 1/8th the area original blue triangle, this is because the
green triangle has a fourth of the length and half the width. The next step is where Archimedes
then uses a leap of logic that surpasses the time. This ingenious step was to generalize the fact
that the subsequent triangles had a factor of 1/8th . For example, the yellow triangles each have
1/8th the area of the previous green triangle, the miniscule red triangles have 1/8th the area of the
previous yellow triangles, and so forth. With this, Archimedes allowed the realm of infinity to
broach the area of Geometry, and changed the idea of summation forever. This could be written
as the following equation –
Fig 1 Fig 2
Math IA - Evolution of Sequences and Series 7 003159-0025
March 23, 2015
Area of Parabolic Segment = 𝑇 + 2 (𝑇
8) + 4 (
T
82) + 8 (
T
83) + ⋯
Where ‘T’ represents the total area of the segment, and each of the terms represents the area of
the colored triangles, the second term associating with the area of the two green triangles, the
next term accounts for the four yellow triangles, etc. This further simplifies to
Area of Parabolic Segment = (1 + (1
4) + (
1
16) + (
1
64) … ) × 𝑇
The following series is an infinite one that exactly determines the ratio of the area of the
parabolic segment to the area of the blue triangle. The next challenge Archimedes faced was to
actually add up this infinite series. This required another leap of logic that used the trademark of
ancient Greece of visual geometry as they did not have Algebra to arrive at an answer.
Archimedes constructed a diagram2 to remarkably deduce the fact that 4
3= (1 + (
1
4) + (
1
16) +
(1
64) … ).
2 Taken from - Proofs without Words: Exercises in Visual Thinking (Classroom Resource Material), by
Roger B. Nelson. ISBN 978-0883857007.
Fig 3
Math IA - Evolution of Sequences and Series 8 003159-0025
March 23, 2015
The above diagram proves that (1
4) + (
1
16) + (
1
64)… sums to 1/3. We can see that a fourth of the
unit square is shaded purple, and then a fourth of a fourth is shaded, and so on. This pictorially
leaves us with the notion that the sum (1
4) + (
1
16) + (
1
64) is one third of the original unit square,
making the summation equal to 1/3. This then allows us to say that 4
3= (1 + (
1
3)) which is true.
This proof revolutionized the idea of summation by incorporating the idea of infinity into the
proof, again, with the use of visual aid in the form of geometry. The notion of infinity becomes
more interesting and complex after the advent of Algebra, which revolutionized culture.
Resurgence of Mathematics in the West (Modern Era)
Advancements in Algebra and the fall of Islam In order to fully grasp the modern era of infinite sequences and series, we must first
understand the rise and fall of Islam, where Algebra had been pioneered. Moving a thousand
years from Ancient Greece, from 800-1100, the Islamic states had been at the forefront of
science and mathematics, so much so that it is even called the Golden Age of Islam. During this
period, the numerals we use today called the Hindu-Arabic numerals were invented and the basis
for Algebra had been established. As seen before, culture does affect mathematics, but it works
the other way around too! This breakthrough in the advent of algebra reformed Islamic culture at
the time as people were immigrating there to witness the paradigm shift in knowledge with
respect to mathematics and in science (especially astrology). Another piece of evidence that
provides insight as to how mathematics shapes culture is the fall of Islam. Hamid al-Ghazali, an
academic scholar at the time, provided a perspective that the manipulation of numbers was the
work of the devil. This ideology took over the people of Islam during 1100, which persuaded
people to forget mathematics and science – causing a whole collapse of a culture and society.
Math IA - Evolution of Sequences and Series 9 003159-0025
March 23, 2015
Convergence, Divergence and Riemann-Zeta Function The advent of algebra marked the first time where the mathematical realm shifted away
from concrete tangible shapes and geometry and moved to a more abstract realm. This new way
of viewing math had to be explored and dealt with by different people and societies, which is
why there was a significant lack of major advancements between 1100 and 1600 in terms of
mathematics. During the turn towards the 1700s however, individuals such as Euler, Gauss and
Cauchy were focused on algebraic and arithmetic definitions of infinite series’. These prominent
mathematicians, making contributions that were beyond their time, noticed that infinite series
came in primarily two different criteria – Convergent and Divergent series. A convergent series
is one where the summation of the terms asymptote to one value or converge at a value. For
example, the infinite series 1 +1
2+
1
4+
1
8+ ⋯ sums to exactly 23.
3 Also could be referred in Appendix – Convergent Table and Graph
Number of Terms Value
1 1
2 1.5
3 1.75
4 1.875
5 1.9375
6 1.96875
Math IA - Evolution of Sequences and Series 10 003159-0025
March 23, 2015
A divergent series is one that does not appear to be approaching to one particular value, growing
bigger and bigger with the addition of every subsequent term. An example of this would be 1 +
2 + 3 + 4 … as one might expect to see a gradual increase to infinity as the number of terms
keeps increasing4.
Number of Terms Value
1 1
2 3
3 6
4 10
5 15
6 21
These mathematicians during this time contributed many methods to deduce whether the series
given is converging or diverging. It might seem impossible to sum a divergent series to other
than infinity but Leonhard Euler (and later Riemann) devised a way to associate a value to the
divergent series, essentially finding an exact number equivalence to the infinite diverging series.
In comparison to the Ancient Greeks, this modern method will have no visual representation and
will be on the basis of algebra – completely juxtaposing how the ancient Greeks viewed and
understood mathematics. This proof will give us an understanding of the fine grained steps and
abstract reasoning the 1700s had to offer in mathematics.
4 Also could be referred in Appendix – Divergent Table and Graph
Math IA - Evolution of Sequences and Series 11 003159-0025
March 23, 2015
Natural Numbers Summation Proof To start the proof, we need to know the expansion of a Taylor function. The Taylor function in
this case is 1
1−x. A Taylor expansion is the representation of a function in terms of an infinite
summation of terms. In this case, the Taylor expansion of 1
1−x is given as
1
1 − x= 1 + 𝑥 + 𝑥2 + 𝑥3 … 𝑥 < 1
Next, we need to derivate both sides, RHS and LHS. We then get -
1
(1 − x)2= 1 + 2𝑥 + 3𝑥2 + 4𝑥3 … 𝑥 < 1
Substituting x for -1 into the equation gives us -
1
4= 1 − 2 + 3 − 4 … 𝑥 < 1
This is the summation notation of the Riemann zeta function, which could also be represented as
Zeta of (s)’ or δ(s) for simplifying the notation.
∑1
ns
∞
n=1
= δ(s)
This summation notation (LHS) can be expanded and rewritten in the (RHS).
∑1
𝑛𝑠
∞
𝑛=1
= 1 +1
2𝑠+
1
3𝑠+
1
4𝑠+ ⋯ − ∞ < 𝑠 < ∞ (𝑒𝑞. 1)
To simplify the notation, we could define this sequence as a function of (s)
𝛿(𝑠) = 1 +1
2𝑠+
1
3𝑠+
1
4𝑠+ ⋯ − ∞ < 𝑠 < ∞ (𝑒𝑞. 1)
To continue the proof, we must multiply both sides by (2−𝑠)
(2−𝑠) 𝛿(𝑠) =1
2𝑠+
1
4𝑠+
1
6𝑠+
1
8𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 2)
We then multiply (𝑒𝑞. 2) by 2 on both sides. Let us call this new function the ‘modified
Riemann-Zeta function’.
Math IA - Evolution of Sequences and Series 12 003159-0025
March 23, 2015
(2 × 2−𝑠) 𝛿(𝑠) =2
2𝑠+
2
4𝑠+
2
6𝑠+
2
8𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 2)
The next step is the most complicated as it sets up the formation of the equation that is needed
for analytical continuation. We take eq. 2 (The modified Riemann-Zeta function) and subtract it
from eq. 1 of the proof (the original Riemann-Zeta function) both in the LHS and RHS. We get
the following equation
(1 − (2 × 2−𝑠)) 𝛿(𝑠) = 1 +1
2𝑠+
1
3𝑠+
1
4𝑠+ ⋯ − (
2
2𝑠+
2
4𝑠+
2
6𝑠+
2
8𝑠 … ) (𝑒𝑞. 3)
We notice something peculiar in the RHS after this step, that every even denominator to the
power (s) will be subtracted. Furthermore, since it is being subtracted by exactly twice the
amount of the original, the sign would change from positive to negative. For example, 2 – 4 = -2
or 2x – 4x = -2. The RHS would now have alternating signs for the even denominators when we
simplify the equation.
(1 − (2 × 2−𝑠)) 𝛿(𝑠) = 1 −1
2𝑠+
1
3𝑠−
1
4𝑠+ ⋯ (𝑒𝑞. 3)
Let us now substitute -1 for (s). We do this to find some sort of correlation between the RHS
and LHS.
(1 − (2 × 2−(−1))) 𝛿(−1) = 1 −1
2−1+
1
3−1−
1
4−1+ ⋯ (𝑒𝑞. 3)
We expand the eq. 3 further, including the original Riemann Zeta function in the LHS 𝛿(−1).
Plugging -1 in eq. 1 reciprocals the terms of the function. We now have the equation
(−3) (1 + 2 + 3 + 4 … ) = 1 − 2 + 3 − 4 … (𝑒𝑞. 4)
Now we can utilize the expanded Taylor series to associate a value to the RHS, which we know
from the derivative of the Taylor expansion is 1
4
(−3) (1 + 2 + 3 + 4 … ) =1
4 (𝑒𝑞. 5)
Finally, bringing the (-3) to the other side leaves us with the famous and counterintuitive
answer that the summation of all the natural numbers is −1
12
(1 + 2 + 3 + 4 … ) =−1
12 (𝑒𝑞. 6)
Math IA - Evolution of Sequences and Series 13 003159-0025
March 23, 2015
Romanticism in Mathematics and Science This remarkable answer was hard to grasp by the people who saw or independently
discovered the proof, Ramanujan himself wrote in his letter to G.H Hardy -
“1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to
me the lunatic asylum as my goal.”5
This completely counterintuitive answer would have been disregarded by the people of Ancient
Greece and that of Islam during the Golden Age. This abstract notion of mathematics could
associate a number with a divergent series, one that we immediately think should have no other
acceptable answer apart from infinity. This answer, although perplexing as it is, was not
immediately thrown out by the likes of Euler, Ramanujan or Riemann for reasons not only due to
their ingenuity in math, but also the cultural ideologies at the time – an idea called Romanticism.
Romanticism is the idea that was prominent in the mid 18th century, a western European cultural
movement. This cultural shift promoted a philosophy called anti-reductionism, which is the idea
that the whole was more valuable than the parts alone6. This directly applies to the field of
infinite summations, especially in the realm of divergent series summations and analytical
continuation. This cultural concept that was growing can be seen in the progress of math, the
answer to the sum of all natural numbers could be more than, intuitively, the sum of its parts
which we take to be infinity. This ideology that dictated the progress of mathematics from 17th
century onwards compared to that of the ancient Greeks starkly contrasts one another. One uses
concrete geometries to arrive at proofs, the other uses abstract reasoning and methods to arrive at
counter-intuitive proofs – however both are equally valid and true, and that’s the beauty of it.
5 Taken from Ramanujan: Letters and Commentary, Srinivasa Ramanujan Aiyangar. Berndt et al. p.53 6 Taken from Molvig, Ole, History of the Modern Sciences in Society lecture course, Sept. 26.
Math IA - Evolution of Sequences and Series 14 003159-0025
March 23, 2015
Another aspect of the Riemann function is its connection to the distribution of the prime
numbers. This is extremely important in number theory, and the connection between prime
numbers and the Zeta function proof is given below.
Prime Product Formula Derivation
We have the Riemann-zeta function below -
𝛿(𝑠) = 1 +1
2𝑠+
1
3𝑠+
1
4𝑠+ ⋯ − ∞ < 𝑠 < ∞ (𝑒𝑞. 1)
Let us multiply 1
2𝑠 on both the sides, let it be called (𝑒𝑞. 2)
𝛿(𝑠)
2𝑠=
1
2𝑠+
1
4𝑠+
1
6𝑠+
1
8𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 2)
We now have to find the difference between equation 1 and equation 2, subtracting LHS and
RHS respectfully gives
𝛿(𝑠) − 𝛿(𝑠)
2𝑠= 1 +
1
3𝑠+
1
5𝑠+
1
7𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 1) − (𝑒𝑞. 2)
The left hand side can be further simplified to
(1 − 1
2𝑠) 𝛿(𝑠) = 1 +
1
3𝑠+
1
5𝑠+
1
7𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 3)
We have in some sense ‘extracted’ (removed) all the terms that are multiples of 2 in the right
hand side and wrote it as a product of the function in the left hand side. This could be explored
more and we could try to extract all multiplies of 3 from the left hand side, by repeating the same
procedure. Let us multiply both sides by 1
3𝑠
(1 − 1
2𝑠)
𝛿(𝑠)
3𝑠=
1
3𝑠+
1
9𝑠+
1
15𝑠+
1
21𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 4)
Math IA - Evolution of Sequences and Series 15 003159-0025
March 23, 2015
We now have to find the difference between equation 3 and equation 4, subtracting LHS and
RHS respectfully and simplifying gives
(1 − 1
2𝑠) (1 −
1
3𝑠) 𝛿(𝑠) = 1 +
1
5𝑠+
1
7𝑠+
1
11𝑠 … − ∞ < 𝑠 < ∞ (𝑒𝑞. 3) − (𝑒𝑞. 4)
Again, we have removed the terms that contain a multiple of 3 in the denominator of the RHS
and defined it as a product in the LHS. The next step of this proof, is to deductively say that as
we keep repeating this process for all the prime numbers we would eventually get the RHS as
one and one the LHS an infinite product of all the primes in terms of ∏ (1 − 1
𝑝𝑠)𝑝 𝑝𝑟𝑖𝑚𝑒 .
∏ (1 − 1
𝑝𝑠)
𝑝 𝑝𝑟𝑖𝑚𝑒
𝛿(𝑠) = 1 (𝑒𝑞. 5)
Eq. 5 can now be simplified further to reach:
𝛿(𝑠) = ∏ (1
1 − 𝑝−𝑠)
𝑝 𝑝𝑟𝑖𝑚𝑒
(𝑒𝑞. 6)
In Conclusion… This proof shows the elegance of modern day arithmetic, it makes connection to other areas
of math that were thought to be completely unrelated – bringing us back full circle to the idea of
prime numbers. Whether it is from Euclid’s definition of primes using lines or Euler’s proof of the
summation of the divergent series of the natural numbers, math is an area of knowledge that is
connected inexorably - linked with not only itself but the nature of reality as a whole. The beauty of
math is that it transcends cultural barriers such as language, social norms and interpersonal relations.
Math itself is dictated by the cultural influences of the time, ranging from the Ancient Greeks’
method of empirically testing math to the modern era of mathematical acceptance of counter-intuitive
answers due to the rise in Romanticism. Math creates and destroys cultures as well, seen from the
rise and fall of Islam. All in all, mathematics not only shapes personal knowledge about universal
truths, but infinitely advances shared cultural knowledge.
Math IA - Evolution of Sequences and Series 16 003159-0025
March 23, 2015
Bibliography
Nelsen, Roger B. Proofs without Words: Exercises in Visual Thinking. Washington,
D.C.: Mathematical Association of America, 1993. Print.
Aiyangar, Srinivasa, and Bruce C. Berndt. Ramanujan: Letters and Commentary.
Providence, R.I.: American Mathematical Society, 1995. Print.
Holmes, Richard the Age of Wonder: The Romantic Generation and the Discovery of the
Beauty and Terror of Science, 2009. Print.
"Sum of Natural Numbers (second Proof and Extra Footage)." YouTube. Ed. Brady
Haran. YouTube. Web. 7 Jan. 2015.
"Euler's Product Formula for the Zeta Function." YouTube. Ed. Exotic Math. YouTube.
Web. 4 Feb. 2015.
Math IA - Evolution of Sequences and Series 17 003159-0025
March 23, 2015
Appendix
Convergent Table and Graph
Infinite summation of 1 +1
2+
1
4+
1
8+ ⋯
Number of Terms Value
1 1
2 1.5
3 1.75
4 1.875
5 1.9375
6 1.96875
Divergent Table and Graph
Infinite summation of 1 + 2 + 3 + 4 …
Number of Terms Value
1 1
2 3
3 6
4 10
5 15
6 21
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7
Summation Value
Number of Terms in the Series
Summation of 1+(1/2)+(1/4)+...
Value
Poly. (Value)
Math IA - Evolution of Sequences and Series 18 003159-0025
March 23, 2015
0
5
10
15
20
25
0 1 2 3 4 5 6 7
Summation Value
Number of Terms in the Series
Summation of 1+2+3+...
Value
Poly. (Value)