Transcript The Harmonic Series

The Harmonic Series
An investigation into the Bernoulli
family or “all families are
psychotic”
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
World History 1600 to 1700
Mayflower lands at
Plymouth Rock
The Great Fire of
London
Captain Vitus Bering
discovers Alaska
Taj Mahal completed
Jamestown, Virginia
established
1600
1620
End of Ming Dynasty
in China
1640
First slaves to North
America
The Relation debuts
1660
1680
1700
The Great Plague of
London
Charles II is
crowned King of
England
1720
United Kingdom of
Great Britain is
formed
Peter the Great
becomes Czar of
Russia
Milton's Paradise
Lost
1740
History of Math 1600 to 1700
• John Napier (1550-1617)
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First to discover natural logarithms
First to use decimal point
Napier’s bones used for multiplication
Magician?
History of Math 1600 to 1700
• Galileo (1564-1642)
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Father of modern observational astronomy
1610, observed the moons of Jupiter
Designed a telescope based on Dutch toy
Discovered phases of Venus, Sunspots, and
improved compass design
History of Math 1600 to 1700
• Johannes Kepler (1571-1630)
▫ Worked with data of Tycho Brahe that described
the motion of planets in the sky
▫ Formulated the laws of planetary motion
▫ Mathematics teachers at a seminary school in
Graz, Austria
History of Math 1600 to 1700
• Pierre de Fermat (1601-1665)
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Father of modern Number Theory
Provided foundation for analytic geometry
Ideas foreshadowed Calculus
Fundamentals of probability theory
Fermat’s Last Theorem
History of Math 1600 to 1700
• Blaise Pascal (1623-1662)
▫ Mathematician, inventor, scientist, and writer
▫ First commercially available mechanical calculator
machine
▫ Barometers, vacuums, and air pressure
▫ Writings on religion, philosophy and ethics
History of Math 1600 to 1700
• Isaac Newton (1642-1727)
▫ I hope you came to the last class
History of Math 1600 to 1700
• Gottfried Wilhelm Leibniz (1646-1716)
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Co-inventor of calculus
Idea of determinants
Infinite sum of reciprocals of triangular numbers
Mechanical calculator
Logic—logical argumentation to symbolic form
History of Calculus
• Johannes Kepler
▫ Volume of wine barrels
▫ New Geometry of Wine Barrels
▫ “At an extrema, a function f(x) is not changing as
tiny changes are made in x.”
History of Calculus
• Pierre de Fermat
▫ Family man, lawyer, mathematics as a pastime
▫ Valued his privacy and published very little
▫ Classical scholar, fluent in Italian, Spanish, Latin,
and Greek
History of Calculus
• Rene Descartes (1596-1650)
▫ Embraced public acclaim and published widely
▫ Too busy to have a wife (my kind of man)
▫ Devoted his whole life to the pursuit of abstract
knowledge
▫ Rarely mentioned the work of contributing
mathematicians (he wanted all the credit)
History of Calculus
• Fermat v. Descartes (1600s?)
▫ First battle was over the refraction of light
▫ Fermat used the ideas of Kepler
▫ In 1637, Fermat published a Method for
Determining Maxima and Minima and Tangents
to Curved Lines
▫ Fermat had calculated the limit of a function and
what is today denoted a first derivative
History of Calculus
• Johann Hudde (1628-1704)
▫ Dutch mathematician who showed how to
differentiate a polynomial of any degree and how
to find its extrema in 1659
History of Calculus
• Leibniz v. Newton (late 1600s)
▫ Newton’s Fluxions were hand-written manuscripts
that only a select few saw
▫ Leibniz made a trip to England, saw Newton’s
manuscripts
▫ 1684, after he returned to Germany, he published
his first paper [insert arbitrarily long title here]
▫ World learned calculus from Leibniz and not
Newton (oh la la!)
▫ 1736, the lost ideas of Newton were published
after his death
History of Calculus
• Fermat v. Descartes (1600s)
▫ First battle was over the refraction of light
▫ Fermat used the ideas of Kepler
▫ In 1629, Fermat published a Method for
Determining Maxima and Minima and Tangents
to Curved Lines
▫ Fermat had calculated the limit of a function and
what is today denoted a first derivative
Origins of the Bernoulli Family
• Originally from Holland and were of Calvinism
religion.
• Fled Holland for Switzerland to avoid Spanish
religious persecutions.
• Nicolaus (1623-1708), father of Jakob, Johann,
and Nicolaus I began a spice business in Basel,
Switzerland.
• The family was not math oriented at all.
Bernouilli Brothers (Jakob)
• Jakob (1654-1705), the eldest son, was pressured
to study theology and philosophy to become a
minister by his parents.
• Graduated from University of Basel with
master’s in philosophy (1671) and licentiate in
theology (1676) but studied math and astronomy
simultaneously.
• Moved to Geneva in 1676 to become a math
tutor and travelled around Europe making
several math correspondences.
Bernoulli Brothers (Jakob)
• Returned to Basel in 1683 where he taught
mechanics and researched math and theoretical
physics.
• Appointed professor of mathematics at the
University of Basel in 1687 and became the chair
in 1695 which he remained until
his death.
Jakob’s Math Contributions
• His Ars Conjectandi, published 8 years after his
death is his chief work. Consisted of 4 sections:
▫ I: Bernoulli trials and distributions and further
advances in expected value
▫ II: first modern writing on combinatorics , the
properties of Bernoulli numbers, and the sum of
powers for interers
▫ III: discussed his probability techniques to games
of chance
▫ IV: Law of large numbers and applied probability
to civil, moral and economic affairs
Bernoulli Brothers (Johann)
• Johann (1667-1748), youngest son of Nicholaus.
• Pressured to study medicine which he began
studying at the University of Basel at the age of 16
but asked his brother to teach him math at the same
time.
• The two began studying and applying Leibniz’s
calculus (considered very obscure math at the time)
• Occupied the chair of mathematics at Groningen
(1695-1705) then succeeded his brother as chair at
Basel after his death (1705-1748)
Johann’s Math Contributions
• Became leader of the Continental
mathematicians after Leibniz’s death in their
battle against the English and was the main
reason why Leibniz’s calculus triumphed over
Newton’s.
• Teacher of the great mathematician Leonard
Euler
Johann and l’Hospital
• During Johann’s time in France (1691-1692) he
taught Guillame Francois Antoine de L’Hopital
calculus.
• Johann was to continue correspondence with
l’Hospital by mailing him his findings for monthly
compensation.
• L’Hospital published the first calculus text, Analyse
des Infinitement Petits in 1696.
▫ Gave credit to Bernoulli brothers for their discoveries
▫ L’Hopital’s rule
Family Feud I
• After two years under Jakob, Johann became his
equal.
• Johann began to brag and belittle his brother.
• Jakob retaliated by calling Johann a parrot.
• The two were constantly publically criticizing
each other, one always trying to trying to one-up
the other.
Challenge of Brachistochrone
• June 1696 Johann published this challenge
problem in Leibniz’s journal and gave until
Easter 1697 to submit a solution.
• 5 solutions were submitted: Johann, Jakob,
l’Hospital, Leibniz and anonymous (Newton)
▫ 4 (excluding l’Hospital’s) appeared in the next
version of the journal.
• Jakob later created and solved a harder version
of the problem in attempt to outdo his brother.
▫ Calculus of variations
Daniel Bernouilli (1700-1782)
• Pressured to study medicine because Johann
claimed there was no money in math.
• Studied medicine at Basel and applied mathematical
physics to it.
• Professor of mathematics in St. Pertersberg (17241733)
• Then returned to Basel and held successive chairs of
medicine, metaphysics and natural phylosophy at
the university.
• Passed away in Basel as a professor of natural
philosophie.
Daniel’s Contributions
• Chief work Hydrodynamique (1738). Arranged
so that all results are consequence of the
conservation of energy.
▫ Used to develop pumps and machines to raise
water
• Several papers of problems connected with
vibrating strings with Leonard Euler.
• Bernouilli principle: as the velocity of a fluid
increases, the pressure decreases
Family Feud II
• In 1734 both Johann and Daniel entered a
contest in the Paris Academy.
• Jointly won the first prize.
• Johann, infuriated, banned Daniel from his
house.
• Johann also stole one of Daniel’s paper and
submitted it with his own name on it.
Bernoulli Family Tree
The Mount Allison Math Olympics
• Problem 1: Find the sum
▫ 1 + 2/3 + 4/9 + 8/27 + 16/81 + …
• Problem 2: Find the sum
▫ 1 + 1/1! + ½! + 1/3! + ¼! + …
• First one to finish wins!
• Note: no Ph.Ds allowed in the competition
The Harmonic Series
The Harmonic Series
The Harmonic Series
Where the rabbit hole led…
The
Harmonic
Series
Harmonic
Numbers
Riemann Zeta
Function
Riemann
Hypothesis
EulerMascheroni
Constant
Digamma
Function
Mertens
Theorem
Harmonic numbers
• A harmonic number comes from truncating the
series
• Another way to express it is using the EulerMascheroni constant and the digamma function
Euler-Mascheroni Constant
=
• γ is used to represent this constant
• Hn is a harmonic number
• It is not known to be irrational (or transcendental
for that matter)
• G.H. Hardy said he would give up his Savillian Chair
at Oxford to anyone who could prove it was
irrational
• It shows up in many integrals
Digamma Function
• A special function derived from the logarithmic
derivative of the gamma function
• Used in a number of number theory applications
Mertens Theorem
=eγ =1.781072…
• Pk denotes the kth prime
• ζ(s) is the Riemann Zeta Function
Riemann Zeta Function
• This generalization of the harmonic series is
known as the Riemann Zeta Function because
Riemann was the one who popularized this
“special function”
• It arises in definite integration
• Has had a profound impact on the prime
number theorem
Riemann Hypothesis
• First published in 1859 paper of Riemann
• States that “the nontrivial Riemann zeta function
zeros all lie on the critical line σ=R[s]=1/2”
• Nontrivial being those values of s other than -2,
-4, -6, … such that ζ(s)=0
• ζ(s) is the Riemann zeta function
• s is a complex variable for which the zeta
function is defined over the complex plane
• R[s] denotes the real part of the of s
Riemann Hypothesis
• One of the problems proposed by the Clay
Mathematics Institute in 2000
• A $1 million prize is attached for those who can
prove it
• Can you?
<http://www.claymath.org/millennium/Rules_
etc/>