calculus-2014-05-21

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DRAFT c○ <strong>2014</strong> Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse normal distribution. To find an approximation you can use the idea of series. In fact, you have to understand complex numbers to really understand the mathematics involved. The answer is given by the error function erf(x), see Figure 5.13. Figure 5.13. Complex Error Function Erf(z). 70
CHAPTER 6 Alternating Harmonic Series As we have already discovered, one of the essential problems of the <strong>calculus</strong> is to determine the area under a curve. A critically important curve to find the area under is y = 1 x as the area under this curve defines the natural logarithm, the inverse of the base e exponential that describes exponential growth. Author’s Note: Did the McLaurin series predate the sum of the alternating harmonic series sum? One would think so or it would be a triviality - in some sense. So what is the history? How was it that Pitero Mengoli discovered it in 1650? In this section we will consider the area that defines ln(2). This is the area under the curve between x = 1 and x = 2 as shown in Figure 6.1. We will try to express this area by approximating it more and more closely via Riemann rectangles. 1 Figure 6.1. Area under the curve y = 1 x for 1 ≤ x ≤ 2.. Copies of the figures below are included in the appendix for you to work with. 1. What is the area enclosed by the square in Figure 6.2? 2. In a copy of Figure 6.3, find and highlight a rectangle whose area is 1 2 . 1 This approach is due to Matt Hudelson (; - ) from “Proof without words: The alternating harmonic series sums to ln(2), Mathematics Magazine, vol. 83, 2010, p. 294. 71
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Discovering the Art of Mathematics
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Contents Acknowledgements v Preface
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Preface: Notes to the Explorer Yes,
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Navigating This Book Before you beg
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CHAPTER 1 What is Area? The simples
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DRAFT c○ 2014 Julian Fleron, Phil
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