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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. 10.1 Early Results.

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Presentation Transcript
Chapter 10Infinite 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

• 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

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 sumof the same series

• proved that harmonic series diverges

• Indian Mathematicians (15th century)

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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

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)

• 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

Taylor’s theorem(Brook Taylor, 1715)

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!

(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 +…

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 + 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

• 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:

Formula sequence

on the other hand:

for all n ≥ 0

Remarks sequence

• 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

10.7 The Zeta Function sequence

• Definition of the Riemann zeta function:

• Euler’s formula:

Remarks sequence

• 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

10.8 Biographical Notes: sequenceGregory and Euler

James Gregory sequenceBorn: 1638 (Drumoak (near Aberdeen), Scotland)Died: 1675 (Edinburgh, Scotland)

• 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)

• He became familiar with methods of Cavalieri

• 1667: “ AndersonVera 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

Leonard Euler was elected to the Royal SocietyBorn: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St. Petersburg, Russia

• 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