Chapter 10 Infinite Series

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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)
• Zenos 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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Oresmes 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
14
14
1/2
14
14
1/2
3/8
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2) Harmonic series diverges
2/4
14
18
1/2
14
1/2
18

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Eulers constant ?
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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 p/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 (1x)
  (integrating of geometric series term-by-term)
• Already known series (such as log (1x) and
  geometric series), Newtons 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 (1x), 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/(t21) )
  can be expanded using geometric series
• For functions of the form (1x)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 ah can be found from
  values at arithmetic sequence a, ab, a2b,...
• First (n1) terms form nth-degree polynomial
  p(ah) whose values at n points coincide with
  values of f(x),i.e. f( akb) p(akb), k 0,
  1, , n-1
• Thus we obtain function f(x) as the limit of its
  interpolation polynomials

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Taylors theorem (Brook Taylor, 1715)
Note Taylors 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 (x1)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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Eulers proof

• Leonard Euler (1707 1783)
• Assume the same is true for infinite polynomial
  equation
• Then
• Therefore

solutions
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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 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!

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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, Fn2 Fn1 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

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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, Fn2 Fn1 Fn for n 0
• f (x) F0 F1 x F2 x2 F3 x3 F4 x4 F5 x5
  0 x x2 2x3 3x4 5x5 8x6 13 x7
  
• We will find explicit formula for f (x)

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• F0 0, F1 1, Fn2 Fn1 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

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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
  thatFn1 / Fn ? (1 v5) / 2 as n ? 8
• 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
• Eulers formula

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Remarks

• Another Eulers result shows that ? (2) p2 /6
• Moreover, Euler proved that? (2n) rational
  multiple of p2n
• Series defining the zeta functionconverges for s
  gt 1and diverges when s 1
• Riemann (1859) considered complex values of s
• Riemann hypothesis (open)if s is a (nontrivial)
  root of ? (s) then Re (s) 1/2

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10.8 Biographical NotesGregory 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 p 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

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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.
  Andrews university
• At St. Andrews Gregory obtained his important
  results on series (including Taylors 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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• Eulers 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