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• 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

32

2

1

2

1

2

1

8

1

4

1

2

11

3

4

4

1

4

1

4

11

64

1

16

1

4

11

32

1|| when 1

32

rr

aararara

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)

22

4

2

3

2

2

2

1432

753tan

7531 xxx

xx 7

1

5

1

3

11

4

and

8

1

7

1

6

1

5

1

4

1

3

1

2

11

Oresme’s proofs2

16

4

8

3

4

2

2

1

2

4

2

3

2

2

2

1432

1)

2

1

2

1

2

11

8

1

8

1

8

1

8

1

4

1

4

1

2

11

8

1

7

1

6

1

5

1

4

1

3

1

2

11

8

1

7

1

6

1

5

1

4

1

3

1

2

11

2) Harmonic series diverges

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

14

14

14

14

14

1/2

14

14

14

14

14

14

1/2

14

= == =

14

14

1/2 18

18

18

…

...

1/2

2/4

3/8

Euler’s constant γ

)1ln(1

4

1

3

1

2

11lim n

nn

577.0

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

0

33

2210

0

33

2210

)()()()(n

nn

n

nn

axaaxcaxcaxcc

xcxcxcxcc

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)

4321

1)1log(

1)()()(1)(1

1

1

1

432

22 33

xxxxdx

xx

xxxxxxxx

dttdtt

x

dtt

x

xx

x

0

2

02

1

02

1

2

1

)1(1

1sin

1

1tan

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

32

!3

)2)(1(

!2

)1(1)1( x

pppx

pppxx p

Gregory-Newton Interpolation formula

)()(3)2(3)3()()()(

)()(2)2()()()(

)()()(

)(!3

21)(

!2

1)()()(

223

2

32

afbafbafbafafbafaf

afbafbafafbafaf

afbafaf

afbh

bh

bh

afbh

bh

afb

hafhaf

• 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

Taylor’s theorem (Brook Taylor, 1715)

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

!3

)(

!2

)()()()( 32 af

haf

hafhafhaf

on so and

)()(

lim

)()()(

lim)(

lim

)(

!3

1)2)((

)(

!2

1)(

)()()(

)(!3

21)(

!2

1)()()(

2

0

00

3

3

2

2

32

afb

af

afb

afbaf

b

afb

afbhbhh

b

afbhh

b

afhafhaf

afbh

bh

bh

afbh

bh

afb

hafhaf

b

bb

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

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)

3

4

4

1

4

1

4

11

32

1

11sum partial n

14

1

4

1

3

1

3

1

2

1

2

11

4

1

3

1

3

1

2

1

2

11

1

11

)1(

1

54

1

43

1

32

1

21

1

th

1

1

n

nn

nn

n

n

1

22222

1

5

1

4

1

3

1

2

11

n n

6

1

5

1

4

1

3

1

2

11

2

122222

n n

Euler’s proof• Leonard Euler (1707 – 1783)

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

• Then

• Therefore

0!7!5!3

1sin 32

xxx

x

x solutions

)0but (

3

22

3

22

21

x

x

x

x

) ofnt (coefficie111

121

xaxxx n

6

1

3

1

2

11

6

1

!3

11

3

1

2

11

2

222

2222

n

n

!7!5!31

sin

!7!5!3sin

32

753

xxx

x

x

xt

ttttt

n

n

nn

n

nn

xxxxa

x

x

x

x

x

xxaxaxa

xxx

xaxaxa

111) ofnt (coefficie

1111

roots - ,,,

01

211

21

221

21

221

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!

xxf )(

01)1(1

1 22

2

xy

xy

• Newton (1671)

• Moreover:

numbers! rational are where 321

3210321

, , r, rr

xaxaxaay rrr

)(

)(

)(

2210

2121110

22

202010010

1

xcxccxb

xcxccxb

xcxccxbay

nnns

n

s

s

n

Puiseux expansion(Victor Puiseux, 1850)

Example

)1(1

)1(

322/12/1

22

xxxxx

xy

xxy

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 +…

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 + 8x6 + 13 x7 + …

• We will find explicit formula for f (x)

• 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

0 0 0

21)(

xx

xxf

Application: general formula for the terms of Fibonacci sequence

partial fractions:

))2/)51((1)()2/)51((1(1)(

2 xx

x

xx

xxf

xxxf

)2/)51((1

1

)2/)51((1

1

5

1)(

geometric series:

2

2

2

2

2

51

2

511

)2/)51((1

1

2

51

2

511

)2/)51((1

1

xxx

xxx

2

22

2

51

2

51

2

51

2

51

5

1)( xxxf

Formula

n

nn

xxxxf2

51

2

51

2

51

2

51

2

51

2

51

5

1)( 2

22

on the other hand:

nn xFxFxFFxf 2

210)(

nn

nF2

51

2

51

5

1

for all n ≥ 0

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

10.7 The Zeta Function• Definition of the Riemann zeta function:

• Euler’s formula:

sss

s4

1

3

1

2

11)(

)(4

1

3

1

2

11

11

1

111

1

1

71

1

1

51

1

1

31

1

1

21

1

1

prime

s

p

sss

pssssss

prime 11

1)(

psp

s

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 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) then Re (s) = 1/2

10.8 Biographical Notes:Gregory and Euler

James GregoryBorn: 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)• 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 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

• 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 EulerBorn: 15 April 1707 in Basel, Switzerland

Died: 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

• 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