Mathematics 153
Notes from June 1, 2020, lecture session

What was covered:
The material from Unit 14, starting with the subheading on infinite series and continuing into the discussion of Bolzano in the early 19th century.   It stopped just short of the material on Cauchy.
Relevant directory files for Unit 14:
Here is a listing of files from Unit 14 which are related to the June 1 lecture, with comments on their significance.  These links are all clickable. 
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14.pdf  This is the main file for the unit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week8/unit11/catenary+parabola.png   This file was previously included, but it was mentioned here in connection with the work of the Bernoullis
https://www.youtube.com/watch?v=LHttKi-QIW8    A YouTube video on the various members of the Bernoulli family, who made numerous contributions to mathematics and physics during the late 17th century and most of the 18th century.    Material from this video (including the sound track) may appear on the final exam.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14e.pdf   This file discusses some of the applications of calculus to questions in algebra and number theory.  The first item is the sequence of Bernoulli numbers, which is named after the first John Bernoulli.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/infinite-series-sum.png  An informal description of the condition for an infinite series to converge.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14a.pdf  An example showing that standard manipulations with infinite series can yield bizarre conclusions if applied without thought to divergent series.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14b.pdf  A much more advanced discussion of the example in the preceding file.  This will not be tested.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/series-sums.pdf  A more detailed discussion of some standard operations involving infinite series which are valid for convergent series but fail for divergent series.
https://www.youtube.com/watch?v=UKHBWzoOKsY    AND
https://www.youtube.com/watch?v=59RIABaGTcs    AND
https://www.youtube.com/watch?v=kwqNDIacqt0    AND
https://www.khanacademy.org/science/electrical-engineering/ee-signals/ee-fourier-series/v/ee-fourier-coefficients-sine    These are Khan Academy videos explaining the theory of trignometric series associated to periodic vibrations, including derivations of the formulas for decomposing such vibrations into a sum of fundamental frequencies and higher order harmonics.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14f.pdf  Examples of waveforms for standard musical instruments, with illustrations and comments on the sizes of various harmonics.   The combined length is about 30 minutes.
https://en.wikipedia.org/wiki/Square_wave  The sounds of a sine wave and approximations to a square wave with higher harmonics added.  (Make sure the sound is turned on!)
https://en.wikipedia.org/wiki/Triangle_wave  Similar file for a triangle wave.
https://en.wikipedia.org/wiki/Sawtooth_wave  Similar file for a sawtooth wave.
https://www.youtube.com/watch?v=H6bgKWgZEKw&list=PLWkr0FoP_-fRT4eyz21QEpejxCZKW5zXt&index=13     AND
https://www.youtube.com/watch?v=JEr6sE1C2kc    Two YouTube videos on infinitesimals from the times of Archimedes, Newton-Leibniz, and the present day (for example, Abraham Robinson's nonstandard analysis).  Material from these videos may appear on a final examination.
http://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch5/lit.html  One phase of Lagrange's work to formulate calculus using infinite series was the following result known as the Lagrange Inversion Theorem:  If  f is a function defined by a convergent power series and it has an inverse function g, then there is a specific formula representing g locally as a convergent power series.   The inversion formula reflects Newton's preferences for dealing with functions.  Something about the inversion theorem might appear on the final exam.