II Towards infinity ¥

Sequences: potential infinity (n) = 1, 2, 3,... (2 n ) = 1, 2, 4, 8,... The reciprocals of the terms of the sequence approach zero. 1/n becomes smaller than every fixed  > 0 But the limit is not assumed: 1/n > 0

Carl Friedrich Gauß ( ) = n = n(n+1)/2 = 5050

Geometric series: 1 + q + q 2 + q q n - (1 + q + q q n-1 + q n )q = 1 - q n q + q q n = The inventor of chess = 2  grains of rice Surface of earth: 5  cm q + q = für IqI < 1 Infinitely many numbers, finite sum:

Nicole d’Oresme ( ) College de Navarre at Paris: Pupil, teacher, chairman Bishop of Lisieux First ideas of analysis Rational powers:4 3 = 64 = 8 2  8 = 4 3/2

times the age of the universe

Not every series converges absolutely: halved and added But we see the same terms!

Gottfried Wilhelm Leibniz ( ) deliberately used the infinite harmoic series: "... so the difference between two harmonic series, may they be infinite though, can be a finite magnitude.“

Francois Viète  ) = Vieta Born and died as a catholic, intermediate Huguenot 1572 Massacre of St. Bartholomew: Huguenots killed Attorney in Fontenay-le-Comte Member of parliament in Rennes and Tours Deciphered the Spanish secret code (500 Zeichen) Greatest French mathematician of the 16. century Theorems about roots of polynomials sin 2  = 2 sin  cos  and further such formulas First infinite product-sequence (1593)

Wallis' product in: Arithmetica Infinitorum (1655) John Wallis ( ) James Gregory ( ) Could calculate all logarithms of positive integers, found the Taylor-series long before Taylor, found 1671 the series of Leibniz, 3 years before Leibniz (1674)

Prism: dispersion of light Hairpin-experiment Reflector telescope Apple tree? 1/r 2 -law of gravitation Mechanics: F = p Isaac Newton ( )

Cambridge Trinity College

Jakob Bernoulli ( ) ax = x 2  x = a /(1+x) = 1 - x + x 2 - x 3 + x ½ = Monk Grandi: God‘s creation from nothing: ½ = = Lemniscate ¥

Leonhard Euler ( ) Pupil of Johann Bernoulli Greatest mathematician of the 18th century. Most productive mathematician of all times His works fill 70 thick books Euler‘s angles (rigid body) Euler‘s gyroscope equations Euler‘s buckling equation Theory of Moon, building of ships, artillery Labelling of triangles e i  = cos  + isin  ln(-1) = ip + ik2p

Friederike von Brandenburg-Schwedt ( ) Popuar book: Letters to a German princess. Translated into 7 languages. The elevation of Berlin is higher than that of Magdeburg, because Spree flows into the Havel and this into Elbe. But far below of Magdeburg!

Tsarina Katharina I. (1684 – ) 2nd wife of Peter the Great Engaged Euler for the Russian academy of sciences

Friedrich II. (the great) ( ) Engaged Euler for the Prussian academy of sciences

Sophie von Anhalt-Zerbst Katherina II (the great) ( )

Zeta function Starting off from the series for sin x he summed the series of inverse squares (Leibniz and the Bernoullis had tried it in vain)-  (2) = 1 + 1/ / / =  2 /6  (4) = 1 + 1/ / / =  4 /90... Euler failed to find the sums for odd exponents. Nobody else succeeded yet.  (-1) = = -1/12

for all x with |x| < 1 for x = 1

The number of primes p < N is about lnlnN Euler found the longest sequence of primes: n(n+1) + 41 supplies primes for n = 0 bis n = 39 41, 43, 47, 53, 61, 71, 83, Euler, like his contemporaries, used divergent sequences for calculations. But he gives the first criterion of convergence: The rest after the infinite term must become infinitely small. Euler always notes the last term, mostly i for numerus infinitus.

= x 0 + x 1 + x 2 + x x i = (-1) 0 + (-1) 1 + (-1) 2 + (-1) = = ½ Leibniz and Jakob B. came to this conclusion too. = = -1 Assumed with Wallis: 1/3 < 1/2 < 1/1 < 1/0 < 1/-1

ln(a/b) = lna - lnb ln2 = ln2  - ln 

ln(a/b) = lna - lnb ln2 = ln2  - ln  /2 - 1/ / 