Math IA - Evolution of Sequences and Series

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


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


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.


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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.


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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.


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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


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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

https://commons.wikimedia.org/wiki/Special:BookSources/9780883857007


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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.


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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


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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.


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


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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’.


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(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)


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.


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)


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.


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.


March 23, 2015

Appendix

Convergent Table and Graph

Infinite summation of 1 +1

2+

1

4+

1

8+ ⋯


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 …


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)


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)


March 23, 2015

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Math IA - Evolution of Sequences and Series