1. Chapter 10 Infinite Series
Early Results
Power Series
An Interpolation on Interpolation
Summation of Series
Fractional Power Series
Generating Functions
The Zeta Function
Biographical Notes: Gregory and Euler

2. Early Results
Greek mathematics: tried to work with finite sums a1 + a2 +…+ an instead of infinite sums a1+ a2 +…+an +… (difference between potential and actual infinity)
Zeno’s paradox is related to
Archimedes: area of the parabolic segment
Both series are special cases of geometric series

3. More examples – series which are not geometric
First examples of infinite series which are not geometric appeared in the Middle Ages (14th century)
Richard Suiseth (Calculator), around 1350:
Nicholas Oresme (1350)
used geometric arguments to find sum of the same series
proved that harmonic series diverges
Indian Mathematicians (15th century)
and

4. 2) Harmonic series diverges
Oresme’s proofs 14 1) .
1/2 14 14 14 1
1/2
1/2
1/2 14 14 14 14 = = = = 1
1/2
1/2
1/2
1/2
1/2 14 14
1/2 14 14
3/8 18
2) Harmonic series diverges
2/4 14 18
1/2
1/2 14 18 …

5. Euler’s constant γ

6. 10.2 Power Series Examples geometric series
series for tan-1 x discovered by Indian mathematicians
Both are expressions of certain function f(x) in terms of powers of x
As the formula for π/4 shows, power series can be applied, in particular, to find sums of numerical series

7. Power series in 17th century
Mercator (published in 1668): log (1+x) (integrating of geometric series term-by-term)
Already known series (such as log (1+x) and geometric series), Newton’s method of series inversion and term-by-term differentiation and integration lead to power series for many other classical functions
Derivatives of many (inverse) transcendental functions (log (1+x), tan -1 x, sin -1 x) are algebraic functions:
Thus method of series inversion and term-by-term integration reduce the question of finding power series to finding such expansions for algebraic functions
Rational algebraic functions (such as 1/(t2+1) ) can be expanded using geometric series
For functions of the form (1+x)p we need binomial theorem discovered by Newton (1665)

8. Binomial Theorem Newton (1665) and Gregory (1670), independently
Note: if p is an integer this is finite sum (polynomial) corresponding to the standard binomial formula
The idea to obtain the theorem was to use interpolation
The Binomial Theorem is based on the Gregory-Newton Interpolation formula

9. Gregory-Newton Interpolation formula
Values of f(x) at any point a+h can be found from values at arithmetic sequence a, a+b, a+2b,...
First (n+1) terms form nth-degree polynomial p(a+h) whose values at n points coincide with values of f(x), i.e. f( a+kb) = p(a+kb), k = 0, 1, … , n-1
Thus we obtain function f(x) as the limit of its interpolation polynomials

10. Taylor’s theorem (Brook Taylor, 1715)
Note: Taylor’s theorem follows from the Gregory-Newton Interpolation formula by letting b → 0

11. 10.3 An Interpolation on Interpolation
In contemporary mathematics interpolation is widely used in numerical methods
However, historically it led to the discovery of the Binomial Theorem and Taylor Theorem
First attempts to use interpolation appeared in ancient times
The first idea of “exact” interpolation (i.e. power series expansion of a given function) is due to Thomas Harriot ( ) and Henry Briggs ( )
Briggs’ “Arithmetica logarithmica” (1624)
Briggs created a number of tables to facilitate calculations
In particular, he was working on such tables for logarithms, introduced by John Napier
One of his achievements was the first instance of the binomial series with fractional p: expansion of (x+1)1/2

12. Summation of Series
Problem of a power series expansion of given function
Alternative problem: finding the sum of given numerical series
Archimedes summation of geometric series
Mengoli (1650)
Another problem:
Attempts were made by Mengoli and Jakob and Johann Bernoulli
Solution was found by Euler (1734)

13. Euler’s proof Leonard Euler (1707 – 1783)
Assume the same is true for infinite “polynomial equation”
Then
Therefore
solutions

14. 10.5 Fractional Power Series
Note: not every function f(x) is expressible in the form of a power series centered at the origin
Example :
Reason: function has branching behaviour at 0 (it is multivalued)
We say that y is an algebraic function of x if p (x,y) = 0 for some polynomial p
In particular, if y can be obtained using arithmetic operations and extractions of roots then it is algebraic, e.g.
The converse is not true: in general, algebraic functions are not expressible in radicals
Nevertheless they possess fractional power series expansions!

15. Puiseux expansion
(Victor Puiseux, 1850)
Newton (1671)
Moreover:

16. Example

17. 10.6 Generating Functions Leonard (Pisano) Fibonacci (1170 – 1250)
Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
Linear recurrence relation
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn for n ≥ 0
Thus F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, F7 = 13 …
What is the general formula for Fn?
The solution was obtained by de Moivre (1730)
He introduced the method of generating function
This method proved to be very important tool in combinatorics, probability and number theory
With a sequence a0, a1, … an,… we can associate generating function f(x) = a0 + a1 x + a2 x2 +…

18. Example: generating function of Fibonacci sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn for n ≥ 0
f (x) = F0 + F1 x + F2 x2 + F3 x3 + F4 x4 + F5 x5 + …= = 0 + x + x2 + 2x3 + 3x4 + 5x5 + 8x x7 + …
We will find explicit formula for f (x)

19. F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn
f (x) = F0 +F1 x + F2 x2 + F3 x3 + F4 x4 + F5 x5 + F6 x6 + …
x f (x) = F0 x + F1 x2 + F2 x3 + F3 x4 + F4 x5 + F5 x6 + …
x2 f (x)= F0 x2 + F1 x3 + F2 x4 + F3 x5 + F4 x6 + …
f (x) – x f (x) – x2 f (x) = f (x) (1 – x – x2 ) =
= F0 +(F1 – F0) x + (F2 – F1 –F0) x2 + (F3 – F2 –F1) x3 + …
f (x) (1 – x – x2 ) = F0 +(F1 – F0) x = x since F0 = 0, F1 = 1

20. Application: general formula for the terms of Fibonacci sequence
partial fractions:
geometric series:

21. Formula
on the other hand:
for all n ≥ 0

22. Remarks
It is easy (using general formula) to show that Fn+1 / Fn → (1 + √5) / 2 as n → ∞
Previous example shows that the function encoding the sequence (i.e. the generating function) can be very simple (not always!) and therefore easily analyzed by methods of calculus
In general, it can be shown that if a sequence satisfies linear recurrence relation then its generating function is rational
The converse is also true, i.e. coefficients of the power series expansion of any rational function satisfy certain linear recurrence relation

23. 10.7 The Zeta Function Definition of the Riemann zeta function:
Euler’s formula:

24. Remarks Another Euler’s result shows that ζ (2) = π2 /6
Moreover, Euler proved that ζ (2n) = rational multiple of π2n
Series defining the zeta function converges for s > 1 and diverges when s = 1
Riemann (1859) considered complex values of s
Riemann hypothesis (open): if s is a (nontrivial) root of ζ (s) then Re (s) = 1/2

25. 10.8 Biographical Notes: Gregory and Euler

26. James Gregory Born: 1638 (Drumoak (near Aberdeen), Scotland) Died: 1675 (Edinburgh, Scotland)

27. Gregory received his early education from his mother, Janet Anderson
She taught James mathematics (geometry)
Note: Gregory's uncle was a pupil of Viète
When James turned 13 his education was taken over by his brother David (who also had mathematical abilities)
Gregory studied Euclid's Elements
Grammar School
Marischal College (Aberdeen)
Gregory invented reflecting telescope (“Optica Promota”, 1663)
In 1664 Gregory went to Italy (1664 – 1668)
University of Padua
He became familiar with methods of Cavalieri

28. 1667: “Vera circuli et hyperbolae quadratura” (“True quadrature of the circle and hyperbola”)
attempt to show that π and e are transcendental (not successful)
first appearance of the concept of convergence (for power series)
distinction between algebraic and transcendental functions
1668: “Geometriae pars universalis” (“A universal method for measuring curved quantities”)
systematization of results in differentiation and integration
the first published proof of the fundamental theorem of calculus

29. During the visit to London on his return from Italy Gregory was elected to the Royal Society
In 1669 Gregory returned to Scotland
He became the Chair of mathematics at St. Andrew’s university
At St. Andrew’s Gregory obtained his important results on series (including Taylor’s theorem)
However, Gregory did not publish these results
He accepted a chair at Edinburgh in 1674

30. Leonard Euler Born: 15 April 1707 in Basel, Switzerland Died: 18 Sept 1783 in St. Petersburg, Russia

31. Euler’s Father, Paul Euler, studied theology at the University of Basel
He attended lectures of Jacob Bernoulli
Leonard received his first education in elementary mathematics from his father.
Later he took private lessons in mathematics
At the age of 13 Leonard entered the University of Basel to study theology
Euler studies were in philosophy and law
Johann Bernoulli was a professor in the University of Basel that time
He advised Euler to study mathematics on his own and also had offered his assistance in case Euler had any difficulties with studying

32. Euler began his study of theology in 1723 but then decided to drop this idea in favor of mathematics
He completed his studies in 1726
Books that Euler read included works by Descartes, Newton, Galileo, Jacob Bernoulli, Taylor and Wallis
He published his first own paper in 1726
It was not easy to continue mathematical career in Switzerland that time
With the help of Daniel and Nicholas Bernoulli Euler had become appointed to the recently established Russian Academy of Science in St. Petersburg
In 1727 Euler left Basel and went to St. Petersburg

33. Euler filled half the pages published by the Academy from 1729 until over 50 years after his death
He made similar contributions to the production of the Berlin Academy between 1746 and 1771
In total, Euler had about 900 published papers
In 1733 Euler became professor of mathematics and the chair of the Department of Geography (at St. Petersburg)
His duties included the preparation of a map of Russia, which could be one of the reason that eventually led to the lost of sight
In 1740 Euler moved in Berlin, where Frederick the Great had just reorganized the Berlin Academy

34. In 1762 Catherine the Great became the ruler of Russia
Euler moved back to St. Petersburg in 1766
Soon after that Euler became completely blind
He dictated his book “Algebra” (1770) to a servant