Lesson 1.pdf - Math 3333 Analysis —– 2018 Summer Shanyu Ji 1 Mathematical Induction Some basic notations Let us denote by Through out this course

Lesson 1.pdf - Math 3333 Analysis —– 2018 Summer Shanyu...

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Math 3333 Analysis —– 2018 Summer Shanyu Ji June 11, 2018 1 Mathematical Induction Some basic notations Through out this course, the following notations will be used: Let us denote by Z , the set of all integers (positive, negative or zero. Sometimes, J can be used too). N = { 1 , 2 , 3 , ... } , the set of all positive numbers (i.e., all the natural numbers). Q := { p q | p, q R , q 6 = 0 } , the set of all rational numbers. R , the set of all real numbers. C := { a + ib | a, b R , i 2 = - 1 } , the set of all complex numbers. We use to denote “the end of a proof.” What is this course going to study? In this course, we are going to retreat Calcu- lus. Our focus will be on studying mathematics using logic and reasoning, instead of just calculation. Why do we need to use logic and reasoning to study mathematics? Several hundreds years ago, after Calculus was discovered by Newton and Leibniz, mathematicians just do calculation, not do proving. However, there was a crack in the foundations of Calculus. For example, from the geometric series, we have 1 1 + x = 1 - x + x 2 - x 3 + ....... 1
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From this, by taking x = 1, Leibniz had suggested that 1 2 = 1 - 1 + 1 + 1 - ..... ; by considering 1 4 = 1 (1+1) 2 , Euler saw 1 4 = 1 - 2 + 3 - 4 + 5 - ..... ; by consider 1 2 = 1 - 1 + 1 + 1 - .... = (1 - 1) + (1 - 1) + (1 - 1) + ... , Grandi referred to the paradoxical result 1 2 = 0 + 0 + ..... This, as suggested by Leibniz, could be compared with the mysteries of the Christian religion that an absolutely infinite force created something out of absolutely nothing. Such mystery lasted for a hundred years. Eventually A.L. Cauchy (1789-1857) took the first step toward unifying the science. He defined “continuity” and “derivative” in terms of the limit, and he gave the first good definition of the limit. Cauchy gave a form of the ( , δ )-definition of limit, in the context of formally defining the derivative, in the 1820’s. The precise ( , δ ) - definition of limit was later formulated by Weierstrass. There was also an interesting story: While attending Cauchys lecture on convergence, P.S. Laplace (1749-1827) became panicked and rushed home. Laplace had just finished his masterpiece that used infinite series as its backbone so that he had to desperately check each one for convergence. Fortunately, all of the infinite series in his books were convergent.
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