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Chapter 10 Infinite Series

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- Early Results
- Power Series
- An Interpolation on Interpolation
- Summation of Series
- Fractional Power Series
- Generating Functions
- The Zeta Function
- Biographical Notes: Gregory and Euler

- Greek mathematics: tried to work with finite sumsa1 + 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

- 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 sumof the same series
- proved that harmonic series diverges

- Indian Mathematicians (15th century)

and

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2) Harmonic series diverges

2/4

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…

- 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

- 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)

- 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 theGregory-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 polynomialp(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

Note: Taylor’s theorem follows from theGregory-Newton Interpolation formula by letting b → 0

- 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 toThomas Harriot (1560-1621) and Henry Briggs (1556-1630)
- 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

- 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)

solutions

- Leonard Euler (1707 – 1783)
- Assume the same is true for infinite “polynomial equation”
- Then
- Therefore

- Note: not every function f(x) is expressible in the form of a power series centered at the origin
- Example :
- Reason: function has branching behaviourat 0(it is multivalued)
- We say that y is analgebraic 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!

Puiseux expansion

(Victor Puiseux, 1850)

- Newton (1671)
- Moreover:

- 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 + Fnfor 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 incombinatorics, probability and number theory
- With a sequence a0, a1, … an,… we can associate generating functionf(x) = a0 + a1 x + a2 x2 +…

- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
- F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fnfor n ≥ 0
- f (x) = F0 + F1 x + F2 x2 + F3 x3 +F4 x4 +F5 x5 +…== 0 + x + x2 + 2x3 +3x4 +5x5 +8x6 + 13x7 +…
- We will find explicit formula for f (x)

0

0

0

- 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 = xsince F0 = 0, F1 = 1

partial fractions:

geometric series:

on the other hand:

for all n ≥ 0

- It is easy (using general formula) to show thatFn+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

- Definition of the Riemann zeta function:
- Euler’s formula:

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

- 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

- 1667: “Vera circuli et hyperbolae quadratura” (“True quadrature of the circle and hyperbola”)
- attempt to show that πand 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

- 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

- 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

- 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

- 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

- 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