Polynomial Approximations and Power Series June 24, 206 Tangent Lines One of the first uses of the derivatives is the determination of the tangent as a linear approximation of a differentiable function f. By the definition of the deriviative and thus f (x 0 ) f(x) f(x 0) x x 0 f(x) f(x 0 ) + f (x 0 )(x x 0 ) = p (x). Exercise. p (x 0 ) = f(x 0 ) and p (x 0 ) = f (x 0 ) 2 Taylor Polynomials The Taylor polynomial of degree 2 agrees at x 0 for f, f, and f (x). Taking derivatives, p 2 (x) = a 0 + a (x x 0 ) + a 2 (x x 0 ) 2 p 2 (x) = a 0 + a (x x 0 ) + a 2 (x x 0 ) 2 p 2 (x 0 ) = a 0 p 2(x) = a + 2a 2 (x x 0 ) p 2 (x 0 ) = a p 2(x) = 2a 2 p 2 (x 0 ) = 2a 2 Thus, a 0 = f(x 0 ), a = f (x 0 ) a 2 = 2 f (x 0 ). Exercise 2. Find the second order Taylor polynomial for sin x, e x, ln(x + ), and x + at x 0 = 0 If we continue, asking for a polynomial n p n (x) = a 0 + a (x x 0 ) + + a n (x x 0 ) n = a j (x x ) ) j
of degree n so that p n and its first n derivatives agree with the corresponding values for f and its iirst n derivatives. Then, for the third derivative and the fourth derivative and, finally, for f (x 0 ) = p n (x 0 ) = 3 2 a 3 a 3 = 3 2 f (x 0 ) = 3! f (x 0 ), f (4) (x 0 ) = p (4) n (x 0 ) = 4 3 2 a 4 a 4 = 4 3 2 f (x 0 ) = 4! f(4)(x 0), f (n) (x 0 ) = p (n) n (x 0 ) = n!a n a n = n! f (n) (x 0 ), Notice that the n + -st derivative of p n (x) is 0. Also, note that the value does not depend on n as long as k n. a k = k! f (k) (x 0 ) Exercise 3. Find the fourth order Taylor polynomial for sin x, ln x, and e x at x 0 = Exercise 4. Find the seventh order Taylor polynomial for sin x and e x at x 0 = 0 Exercise 5. Approximate sin π/3 using its seventh order Taylor polynomial. Then, We do not need to know a function explicitly to compute its Taylor polynomial. For example, if y = x 2y, y(0) =. () y = x 2y, y (0) = 0 2y(0) = 2 y = 2y y (0) = 2y (0) = 2( 2) = 5 y = 2y y (0) = 2y (0) = 0 y = 2y y (0) = 2y (0) = 20 and the four order polynomial for y at x 0 = 0 is + 2x 5 2 x2 5 3 x3 + 5 6 x4. Based on the generalized mean value theorem, careful analysis of the Taylor polynomial shows that we can obtain a bound on the error E n (x) between of the difference between p n (x) and f(x), namely, E n (x) = f(x) p n (x) M (n + )! x x 0 n+, where M = max f (n+) (x) on the interval between x 0 and x. Exercise 6. Find the solution to () and then find the fourth order polynomial for y at x 0 = 0 using this solution. Exercise 7. Give an estimate on the error term for p n (x) for f(x) = sin x. 2
3 Power Series The next question is: Can we continue this approximation indefinitely? Is lim n n f (j) (x 0 ) (x x 0 ) j = j! If so, for what values of x does the limit converge. First, we have to investigate the question: When does the limit: lim n exist? Again, if so, for what values of x does the limit converge. We say that a power series (2) converges absolutely if f (j) (x 0 ) (x x 0 ) j fy(x)? j! n a j (x x ) ) j = a j (x x 0 ) j (2) a j (x x 0 ) j converges. If the limit fails to exist, we say that the power series diverges. 3. Radius of Convergence A power series will converge for some values of the variable x and may diverge for others. All power series converge at x = x 0. There is always a number ρ with 0 ρ such that the series converges absolutely whenever x x 0 < ρ and diverges whenever x x 0 > ρ. This number is called the radius of convergence. The interval (x 0 ρ, x 0 + ρ) is called the interval of convergence. The series may or may not converge for x x 0 = ρ. If (2) converges only for x = x 0, then ρ = 0. If it converges for all x, then we say that ρ =. Exercise 8. The geometric series ax j has radius of convergence ρ =. In this case, show that the infinite sum is What happens for x =? a x. The simplest test for a comparison test If (2) converges absolutely for x (x 0 ρ, x 0 + ρ) and if for some number J, b j < a j for all j > J, then b j (x x 0 ) j We have two tests for convergence based on comparisons with the geometric. 3
Ratio Test If the limit L = lim a j + j a j exists, then in (2), the radius of convergence of (2) ρ = L. Root Test If the limit l = lim j a j j exists, then in (2), the radius of convergence of ρ = /l. (Here, /0 =.) Exercise 9. Find the radius of convergence for j 2 x j, 2 j j (x )j, and j! xj 3.2 Arithmetic Operations Let f and g be two power series converging in a interval about x 0. f(x) = g(x) = a j (x x 0 ) j b j (x x 0 ) j Addition and Subtraction The power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. f(x) ± g(x) = (a j ± b j )(x x ) ) j. where Multiplication The power series for the product is sometimes called the Cauchy product. f(x)g(x) = c j = c j (x x 0 ) j j a i b j i = (a b) j. This is is known as the convolution of the sequences a j and b j. i=0 4
3.3 Uniqueness of Power Series We begin with the following fact, which is somewhat technical to prove. If a power series is equal to 0 is some interval then each of the coefficients a j is 0. In particular, if two power series b j (x x 0 ) j and c j (x x 0 ) j agree for x in some interval, then the difference 0 = b j (x x 0 ) j c j (x x 0 ) j = (b j c j )(x x 0 ) j and so b j = c j for all all j. 3.4 Differentiation and Integration Power series are differentiable on the domain of convergence. So, for x (x 0 ρ, x 0 + ρ) Differentiation Termwise differentiation gives the power series for the derivative. f (x) = ja j (x x 0 ) j Because a j+ a j and f has the same radius of convergence as f. j= ρ, we have (j + )a j+ ja j ρ Integration Termwise integration gives the power series for the integral. and f has the same radius of convergence as f. a j f(x) dx = j + (x x 0) j+ + c a j+ a j ρ, we have a j+ /(j + 2 a j /(j + ) ρ Exercise 0. Use the fact that the geometric series x j = x. to determine the power series for ( x) 2 + x ln( + x) 5 + x 2 tan (x).
Exercise. Show that a j x j = j=j 0 a j l x j l j=j 0+l 3.5 Analytic Functions A function f is called analytic at x 0 is f can be represented as a power series centered at x 0 with a positive radius of convergence. f(x) = a n (x x 0 ) j, Properties of Analytic Functions x x 0 < ρ Polynomials are analytic for every value x 0. If f and g are analytic at x 0 so are f + g, fg, and f/g provided that g(x 0 ) 0. All the derivatives of f are analytic with the same interval of convergence as f. The antiderivative of f is analytic with the same interval of convergence as f. The terms a j = f (j) (x 0 ) j! Thus, the power series is the limit of the Taylor polynomials. Any power seriesregardless of how it is derived? function is the Taylor series of that function. that converges in some neighborhood of x 0 to a Exercise 2. Explain why (x x 0 ) p is not analytic at x 0 if p < 0 or if p > 0 and not an integer. 6