Section 10.8 Reading Assignment: Taylor and Maclaurin Series

Answer Only Exercise 1, 2, and 3 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for and what is really about. Please be very careful with this assignment.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Section 10.8 Reading Assignment: Taylor and Maclaurin Series Instructions. Read through this assignment and complete the three exercises below by reading the appropriate passages of the textbook. Now that we introduced power series, one question that might arise is if we start off with a function, can we find a power series representation for it? The answer is yes, if the function is infinitely differentiable and we can compute all of its derivatives at a point. Exercise 1. Read the subsection "Series Representations" (p. 636-637). Explain the process by which the formula for the coefficients of the power series an = (n) (@) is obtained. Be sure to n! include an explanation for where the factorial comes from. Exercise 2. Read Example 2 (p. 638). Explain the pattern behind the derivatives of e* at x = 0 and how this translates to the Maclaurin series of e*. Exercise 3. Read Example 3 (p. 639). Explain the pattern behind the derivatives of cos(x) and how this translates to the Maclaurin series of cos(x) when written in Sigma notation. Notice the comment right after the Definition on p. 638. Notice that the order of the Taylor polynomial refers to the highest derivative used in the formula. It does not refer to the degree, and more importantly it does not refer to the number of terms being added. That last point does come up. For example, the 4th order Taylor polynomial of cos(x) is 1 - - + -, which only has three terms. Keep this 2! in mind when dealing with the order of a Taylor polynomial.The Taylor formula is useful, but not as useful as these examples would have you believe. In these cases, the function is easy enough that there's a pattern to the derivatives. We can also compute the Taylor series for polynomials, where at high enough order, the derivatives will become zero. But anything more complicated, we will require infinitely many complicated derivative computations. More useful is to use the power series operations that we discussed in the previous reading to take fundamental power series and build up power series for more complicated functions. It's important to note that the formula is not a silver bullet for finding power series; sometimes we have to use power series manipulation methods. Technically, more is required than this. Example 4 (p. 639 - 640) gives an example where the function is infinitely differentiable but whose associated power series is 0 instead of the function. We will treat this as a pathological case though and ignore it. This power series representation works in the cases we care about.Chapter 10 Infinite Sequences and Series 10.8 Taylor and Maclaurin Series We have seen how geometric series can be used to generate a power series for functions such as f(x) = 1/(1 - x) or g(x) = 3/(x - 2). Now we expand our capability to repre- sent a function with a power series. This section shows how functions that are infinitely differentiable generate power series called Taylor series. In many cases, these series pro- vide useful polynomial approximations of the original functions. Because approximation by polynomials is extremely useful to both mathematicians and scientists, Taylor series are an important application of the theory of infinite series. Series Representations We know from Theorem 21 that within its interval of convergence / the sum of a power series is a continuous function with derivatives of all orders. But what about the other way around? If a function f(x) has derivatives of all orders on an interval, can it be expressed as a power series on at least part of that interval? And if it can, what are its coefficients? We can answer the last question readily if we assume that f(x) is the sum of a power series about x = a, f(x) = _a(x - a)" = an taj(x - a) + a,(x - a) + . .. + a (x - a) + ... with a positive radius of convergence. By repeated term-by-term differentiation within the interval of convergence /. we obtain f' (x) = a, + 2az(x - a) + bay(x - a] + .. + na(x - a)"-1+.... f"(x) = 1 . 2a2 + 2 - 3a;(x - a) + 3 - 4a,(x - a) + .... f"(x) = 1 .2 . 3a, + 2 .3 . 4a(x - a) + 3 .4 . 5a;(x - a)+ .... with the nth derivative being f"(x) = nla, + a sum of terms with (x - a) as a factor. Since these equations all hold at x = a, we have (a) = 1 - 202. (a) = 1.2.303. and, in general, ful(a) = nla.Chapter 10 Infinite Sequences and Series 10.8 Taylor and Maclaurin Series 637 These formulas reveal a pattern in the coefficients of any power series 2,-,a,(x - a)" that converges to the values of f on / ("represents f on /"). If there is such a series (still an open question), then there is only one such series, and its ath coefficient is MY(a) If f has a series representation, then the series must be f(x) = f(a) + Flailx - a) + 1 (@), 2! (x - a) + . .. ( x - apt .... (1) But if we start with an arbitrary function f that is infinitely differentiable on an interval containing x = a and use it to generate the series in Equation (1), does the series converge to f(x) at each x in the interval of convergence? The answer is maybe-for some functions it will but for other functions it will not (as we will see in Example 4). HISTORICAL BIOGRAPHIES Taylor and Maclaurin Series Brook Taylor The series on the right-hand side of Equation (1) is the most important and useful series (1685-1731) we will study in this chapter. www. goo . g1/5450x1 Colin Maclaurin (1698-1746) www. goo . g1/vLZONQ DEFINITIONS Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by fat x = a is ( 4 ( - a = fla) + falx - a) + a) + f(a ) 2! ( - 0) + ... d ( - apt.... The Maclaurin series of f is the Taylor series generated by fat x = 0, or PR (@) *= f0)+ fox + f(0) 21+... + n!The Maclaurin series generated by f is often just called the Taylor series of f. EXAMPLE 1 Find the Taylor series generated by f(x) = 1/x at a - 2. Where, if anywhere, does the series converge to 1/x7 Solution We need to find f(2), f(2), f(2). ... . Taking derivatives we get f(x) = x, f(x) = -x' f (x) = 23..., f"(x) = (-1)"nix ("+1) so that ((2) = 2 =; f(2)=-1 f"(2) f (2) (-1) 2! The Taylor series is f(2) + f'(2)(x - 2) - f (2) 21 ( x - 2)- + . .. + f" (2) n! (x - 2)" + . . . = (x - 2) (x - 2)2 N- + (x - 2)" 22 23 +....Chapter 10 Infinite Sequences and Series 638 Chapter 10 Infinite Sequences and Series This is a geometric series with first term 1/2 and ratio r = -(x - 2)/2. It converges absolutely for | x - 2|