I NEED TASK 5 TO BE WORKED OUT ONLY.

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.. . . . .. '-1 '2 '3 Section 116. Sincej IS Infinitelylarge, JT 2 1,..., JT 2 1. JT jl 1 jZ 1 j3 1 = 1, and so forth. It follows that 23, _ 2, 33 _ 3, 43 _ 4, and so forth. When we substitute 2 2 3 3 4 4 these values [into equation (2)], we obtain 1 + g + l: .22 + 12223 + 12% + This equation expresses a relationship between the numbers a and k, since when we let 2 : 1, wehave i1+k+ k2 + k3 + (3 '1' 1 1-2 1-2-3 ) Section 122. Since we are free to choose the base a. for the system of logarithms, we now choose a in such a way that k : 1. Then the series found above in Section 116. 1+1+4L+ 1 1 1-2 1-2-3 +... (4) is equal to a. If the terms are represented as decimal fractions and summed. we obtain the value a = 2.71828182845904523536028.... When this base is chosen, the logarithms are called natural or hyperbolic. The latter name is used since the quadrature of a hyper bola2 can be expressed through these logarithms. For the sake of brevity for this number 2718281828459 . .. we will use the symbol e, which will denote the base for the natural or hyperbolic logarithms. Task 5 Why do you think Euler chose a \"in such a way that k = 1\" in his series (3)? We can obtain an expression for Euler's special a value as the limit of a sequence, and then use modern methods with Euler's ideas to prove this sequence converges. To justify Euler's work from a modern point of View, let's look at the key equation (2) loDp;+louom 1333 1-2-3 1-2j-3j 2+ - 1 (1 Mm)? = 1 + ikz + and set A". = 1,2 = 1 as Euler did, but suppose j is a natural number"