I JUST NEED TASK 4

Euler's Rediscovery of e David Ruch* May 9, 2018 1 Introduction The famous constant e is used in countless applications across many fields of mathematics, and resurfaces periodically in the evolution of mathematics. In 1683, Jacob Bernoulli essentially found e while studying compound interest and evaluating the sequence (1 + 1/j) as j - co. By 1697, his brother Johann Bernoulli was working with the calculus of exponentials [Bernoulli, 1697). However, a full understanding was missing. The connection between logarithms and exponential functions was still not well understood, and mathematicians couldn't agree on how to define logarithms of negative numbers. Leonhard Euler would later clear up the confusion on logarithms of negative numbers, and clarify the idea of a logarithmic function [Euler, 1749]. In 1748, Euler published one of his most influential works, Introduction Analysin Infinitorum [Euler, 1748]. This was translated into English by John Blanton as Introduction to Analysis of the Infinite [Blanton, 1988] and we shall quote his translation with a few minor changes. In Chapter VI, Euler discussed logarithms for various bases and their properties. Logarithms were well known in Euler's day, and tables of common logarithms (base 10) had been compiled, as no scientific calculators were available in 1748. Euler examined exponential and logarithmic functions in Chapter VII, especially as infinite series. We are particularly interested in how e appears naturally in his development of these functions. 2 Euler's Definition of e Part of Euler's challenge in working with logarithmic functions was to find a logarithmic base a for which infinite series expansions are convenient. It is here that Euler derived e, both as the limiting value of (1 + 1/j) and as the infinite series 1 + + + 1 1. 2 1.2.3 1.2 .3. 4 + . As was common in his day, Euler worked with infinitely small and large numbers, a practice that has largely been abandoned with the modern definition of limit. Nevertheless, Euler used his infinitely small and large numbers with great skill, as we shall see.Section 114. Since a" = 1, when the exponent on (1 increases, the power itself increases, provided a is greater than 1. It follows that if the exponent is infinitely small and positive, then the power also exceeds 1 by an infinitely small number. Let (.1) be an infinitely small number, a" = 1 +1!) where "(1] is also an infinitely small number. we let 1!) : kw. Then we have a" : 1 + km, and with a as the base for logarithms, we have m : log (1 + kw). EXAMPLE In order that it may be clearer how the number k depends on a , let a = 10. From the table of common logarithms.1 we look for the logarithm of a number which exceeds 1 by the smallest 1 1 1000000 ' 1000000 ' possible amount, for instance, 1 + so that law 2 Then log (1 + 1 ) = log 1000001 = 000000043429 = (.0. Since kw = 000000100000, it 1000000 1000000 1 _ 43429 _ 100000 _ . . . , follows that E 100000 and is 43429 2.30258. We see that it IS a finite number which depends on the value of the base 0,. If a different base had been chosen, then the logarithm of the same number 1 + law will differ from the logarithm already given. It follows that a different value of is will result. Task 1 To gain some Visual insight into what Euler was doing, plot y = a" and y : 1 l few for a : 10 and k = 1,1030%? 3 2.30258. Euler claimed these quantities of" and 1 + kw should be identical for "infinitely small" 00. Would changing the is value to something else, say *3, change anything about your plot and this claim? Task 2 Use Euler's ideas and a scientic calculator to estimate is for a = 2. Get a Visual check by plotting y = 2"J and y = 1 + kw together. Euler was interested in nding an a value for which exponential and logarithmic expansions are nice and easy to work with. He derived a series expansion in his Section 115. Section 115. Since a" = 1 + 1, we have a/w = (1 + )' , whatever value we assign to j. It follows that 1 . 2 aww = 1 + 2 kw + j ( j - 12 k 2 w2 + j ( j - 1) ( j - 2x3w3 + ... 1 . 2 . 3 (1) If now we let j = 3, where z denotes any finite number, since w is infinitely small, then j is infinitely large. Then we have w = , where w is represented by a fraction with an infinite denominator, so that w is infinitely small, as it should be. When we substitute & for w then at = (1+ kz/j)' = 1+ -kz+1(3-12k2,2 1(j-1)( j-2)13,3 1 . 2 . j 1 . 2j . 3j + : 1 ( j - 1) ( j -2) ( j - 3 4 2 4 + .... 1 . 2j . 3j . 4j (2) We would like to capture the spirit of Euler's ideas but put his work on modern foundations by avoiding infinitely small and large numbers. Task 3 Assume a > 1 and w is a small, positive finite number defined by a" = 1 + y and k = v/w. (a) What theorem was Euler using to obtain (1)? For what y values is this series known to converge? (b) Verify the algebraic details needed to obtain (1) from this theorem. Task 4 Assume a > 1 and w is a small, positive finite number defined by a" = 1 + y and k = v/w and j = z/W. (a) What is the general nth term in the series (2)? (b) Verify the algebraic details needed to obtain (2) from j = z/w and (1). Euler next used his infinitely large numbers to produce an infinite series expression for his ideal logarithm base a. At this point in his book, Euler set z = 1 to find his special value for a