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Unformatted text preview: Infinite Series Tom Rike Berkeley Math Circle March 24, 2002 1 Background Although the rigor of this talk will not be stringent enough for mathematicians, the material is quite interesting and the techniques quite useful. I do not know infinite series well enough to say that I truly understand the topic, but I enjoy investigating it in the dozens of books on infinite series in my library and revelling in the amazing results. This ambivalent feeling about infinite series probably stems from the fact that I did not understand anything about infinite series in my calculus class 37 years ago and the other fact that I must now teach infinite series every year to my students. One of the highlights of the year (at least for me) in my class, after the Advanced Placement exam in May, is to see how the 28 year old Euler in 1735 solved a problem known as the Basel Problem. This problem had been proposed by Jakob Bernoulli in 1689 when he collected the all of the work on infinite series of the 17 th century in a volume entitled Tractatus De Seriebus Infinitis . Bernoullis comment in this volume was that the evaluation is more difficult than one would expect. Little did he know how difficult it really was and that it would take almost 50 years for the problem to be solved. In my class, we then see how the problem can be solved to the satisfaction of mathematicians today, using only knowledge gained in first year calculus. This problem is highlighted in the first volume of Polyas Mathematics and Plausible Reasoning, Volume 1  in his discussion on the use of analogy leading to discovery. It is this method of solving the problem that this talk will address. The real proofs using elementary calculus are many, but not appropriate for this circle. If you have studied calculus then you will find proofs and references to proofs in  ,  , and an extensive bibliography is given by E. L. Stark in Mathematics Magazine , 47 (1974) pp 197-202. A word of caution is in order about entering fields without a proper foundation. It is easy to do things by example and never attain an understanding. This is the issue of How versus Why that Paul Zeitz gave a talk on last year. It is interesting to see the remarks of G. Chrys- tal in Textbook of Algebra  written in 1889. A practice has sprung up of late (encouraged by demands for premature knowledge in certain examinations) of hurrying young students into the manipulation of the machinery of the Differential and Integral Calculus before they have grasped the preliminary notions of a Limit and of an Infinite Series , on which all the meaning and all the uses of the Infinitesimal Calculus are based. Besides being a sham, this is a sin against the spirit of mathematical progress. It is remarkable to see over one hundred years later that we are still fighting this battle. On the other hand, it is never too soon to begin thinking about big ideas and seeing how the great minds approached them. In this webegin thinking about big ideas and seeing how the great minds approached them....
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