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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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Chapter 10 Infinite Series

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Chapter 10 infinite series l.jpg

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


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10.1 Early Results

  • 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


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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)

and


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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

18

14

1/2

1/2

14

18


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Euler’s constant γ


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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


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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)


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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 theGregory-Newton Interpolation formula


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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 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


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Taylor’s theorem(Brook Taylor, 1715)

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


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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 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


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10.4 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)


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Euler’s proof

solutions

  • Leonard Euler (1707 – 1783)

  • Assume the same is true for infinite “polynomial equation”

  • Then

  • Therefore


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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 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!


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Puiseux expansion

(Victor Puiseux, 1850)

  • Newton (1671)

  • Moreover:


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Example


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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 + 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 +…


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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)


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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


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Application: general formula for the terms of Fibonacci sequence

partial fractions:

geometric series:


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Formula

on the other hand:

for all n ≥ 0


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Remarks

  • 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


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10.7 The Zeta Function

  • Definition of the Riemann zeta function:

  • Euler’s formula:


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Remarks

  • 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


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10.8 Biographical Notes:Gregory and Euler


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


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  • 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


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  • 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


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  • 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


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Leonard EulerBorn: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St. Petersburg, Russia


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  • 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


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  • 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


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  • 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


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  • 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


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