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The Harmonic Series

The Harmonic Series. An investigation into the Bernoulli family or “all families are psychotic”. History of Math 1600 to 1700. John Napier (1550-1617) First to discover natural logarithms First to use decimal point Napier’s bones used for multiplication Magician?.

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The Harmonic Series

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  1. The Harmonic Series An investigation into the Bernoulli family or “all families are psychotic”

  2. History of Math 1600 to 1700 • John Napier (1550-1617) • First to discover natural logarithms • First to use decimal point • Napier’s bones used for multiplication • Magician?

  3. History of Math 1600 to 1700 • Galileo (1564-1642) • 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

  4. 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

  5. History of Math 1600 to 1700 • Pierre de Fermat (1601-1665) • Father of modern Number Theory • Provided foundation for analytic geometry • Ideas foreshadowed Calculus • Fundamentals of probability theory • Fermat’s Last Theorem

  6. 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

  7. History of Math 1600 to 1700 • Isaac Newton (1642-1727) • I hope you came to the last class

  8. History of Math 1600 to 1700 • Gottfried Wilhelm Leibniz (1646-1716) • Co-inventor of calculus • Idea of determinants • Infinite sum of reciprocals of triangular numbers • Mechanical calculator • Logic—logical argumentation to symbolic form

  9. 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.”

  10. 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

  11. 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)

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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.

  17. 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.

  18. 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.

  19. Jakob’s Math Contributions • His ArsConjectandi, 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

  20. 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)

  21. 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

  22. 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 InfinitementPetits in 1696. • Gave credit to Bernoulli brothers for their discoveries • L’Hopital’s rule

  23. 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.

  24. 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

  25. 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 (1724-1733) • 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.

  26. 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

  27. 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.

  28. Bernoulli Family Tree

  29. 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

  30. The Harmonic Series

  31. The Harmonic Series

  32. The Harmonic Series

  33. Where the rabbit hole led…

  34. Harmonic numbers • A harmonic number comes from truncating the series • Another way to express it is using the Euler-Mascheroni constant and the digamma function

  35. 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 =

  36. Digamma Function • A special function derived from the logarithmic derivative of the gamma function • Used in a number of number theory applications

  37. Mertens Theorem • Pk denotes the kth prime • ζ(s) is the Riemann Zeta Function =eγ=1.781072…

  38. 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

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