Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series

Transcription

1 .... Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American March 4, 20 K.Maruno (UT-Pan American) Analysis II March 4, 20 / 8

2 Power Series Definition Let a be a fixed real number. A power series in x a (centered at a, about a) is an infinite series of the form c n (x a) n where {c n } is a sequence and x is a real number. K.Maruno (UT-Pan American) Analysis II March 4, 20 2 / 8

3 Power Series Let Theorem c n (x a) n be a power series. Let L = lim sup n a n n. Let 0 if L = R = if 0 < L < L if L = 0 Then the power series c n (x a) n converges absolutely if x a < R and diverges if x a > R. The value R is called the radius of convergence of the power series c n (x a) n. K.Maruno (UT-Pan American) Analysis II March 4, 20 3 / 8

4 Root Test Theorem: Root Test Let {a n } be a sequence and let L = lim sup n a n /n (L = is allowed). (i) If L <, then a n converges absolutely. (ii) If L >, then n= a n diverges. n= The root test gives no information if lim sup n a n /n =. K.Maruno (UT-Pan American) Analysis II March 4, 20 4 / 8

5 Ratio Test Theorem: Ratio Test Let a n be a series such that n= L = lim n a n+ a n exists (L = is allowed). (i) If L <, then a n converges absolutely. (ii) If L >, then n= a n diverges. n= The ratio test gives no information if lim n a n+ a n =. K.Maruno (UT-Pan American) Analysis II March 4, 20 5 / 8

6 Power Series (Example ) n! (x )n = + (x ) + 2! (x )2 + 3! (x ) a n = n! (x )n. Ratio test lim a n+ n a = lim (x )n+ (n+)! n n (x )n = lim x n n + = n! lim x = 0 <. So this series is convergent for all x. The radius n n + of convergence is R =, the interval of convergence is (, ). K.Maruno (UT-Pan American) Analysis II March 4, 20 6 / 8

7 Power Series (Example 2) n (x )n = + (x ) + 2 (x )2 + 3 (x ) a n = n (x )n. Ratio test lim a n+ n a = lim (x )n+ (n+) n n (x )n = lim n n (x ) n + = n n lim x = x. So this series is convergent for x <, n n + i.e., 0 < x < 2. For x = 2, this series is. This is divergent. n ( ) n For x = 0, this series is. This is convergent. n So the radius of convergence is R =, the interval of convergence is [0, 2). K.Maruno (UT-Pan American) Analysis II March 4, 20 7 / 8

8 Exponential Function exp(x) a n = n! xn. Ratio test n! xn = + x + 2! x2 + 3! x lim n a n+ a n = lim n = lim n n + x = 0. (n+)! xn+ n! xn = lim n n + x So the radius of convergence is R =, the interval of convergence is (, ). K.Maruno (UT-Pan American) Analysis II March 4, 20 8 / 8

9 Sine Function sin x ( ) n (2n + )! x2n+ = x x3 3! + x5 5! x7 7! +... a n = ( ) n (2n+)! x2n+. Ratio test lim a n+ a = lim n n ( ) n = lim n (2n + 2)(2n + 3) x2 n ( ) n+ (2n+3)! x2n+3 (2n+)! x2n+ = lim n (2n + 2)(2n + 3) x2 = 0. So the radius of convergence is R =, the interval of convergence is (, ). K.Maruno (UT-Pan American) Analysis II March 4, 20 9 / 8

10 Cosine Function cos x ( ) n (2n)! x2n = x2 2! + x4 4! x6 6! +... a n = ( ) n (2n)! x2n. Ratio test lim a n+ n a = lim ( ) n+ n n ( ) n = lim n (2n + )(2n + 2) x2 = lim n (2n + )(2n + 2) x2 = 0. (2n+2)! x2n+2 (2n)! x2n So the radius of convergence is R =, the interval of convergence is (, ). K.Maruno (UT-Pan American) Analysis II March 4, 20 0 / 8

11 Differentiation Definition: Derivatives Let f be a real-valued function defined at all points of an open interbal (a, b) in R. For each x in (a, b), we define f (x) = lim y x f(y) f(x) y x if this limit exists. If f (x) is defined, we say f is differentiable at x, and we call the number f (x) the derivative of f at x. If f (x) is not defined, we say f is not differentiable at x. If f is differentiable at each point of (a, b), we say that f is differentiable on (a, b). Class C n A real-valued function f is said to be of class C n on (a, b) if f (n) (x) exists and is continuous for all x in (a, b). If f is of class C n for all positive integer n, then f is said to be of class C. K.Maruno (UT-Pan American) Analysis II March 4, 20 / 8

12 Power Series Representations A funtion f(x) Find a power series representation of f(x): A power Series f(x) = c 0 + c (x a) + c 2 (x a) 2 + c 3 (x a) 3 + f(a) = c 0 f (a) = c f (a) = 2c 2 f (a) = 3!c 3... So c n = f (n) (a) n! K.Maruno (UT-Pan American) Analysis II March 4, 20 2 / 8

13 Taylor Series Taylor Series Taylor series of the function f at a (about a or centered at a) f(x) = f (n) (a) (x a) n n! = f(a) + f (a)(x a) + f (a) 2! Maclaurin Series f(x) = Maclaurin Series f (n) (0) n! x n = f(0) + f (0)x + f (0) 2! (x a) 2 + f (a) (x a) 3 + 3! x 2 + f (0) x 3 + 3! K.Maruno (UT-Pan American) Analysis II March 4, 20 3 / 8

14 Taylor Series (Examples) Maclaurin series of exp(x) exp(x) = n! xn = + x + 2! x2 + 3! x Maclaurin series of sin x sin x = ( ) n (2n + )! x2n+ = x x3 3! + x5 5! x7 7! +... Maclaurin series of cos x cos x = ( ) n (2n)! x2n = x2 2! + x4 4! x6 6! +... Taylor series of exp(x) at 2 exp(x) = f (n) (2) (x 2) n = n! e 2 (x 2)n n! = + e 2 (x 2) + e2 2! (x 2)2 + e2 3! (x 2) K.Maruno (UT-Pan American) Analysis II March 4, 20 4 / 8

15 Taylor Polynomials nth-degree Taylor polynomial of f(x) at a Note T n (x) = n f (i) (a) (x a) i i! i=0 = f(a) + f (a)(x a) + f (a) 2! + + f (n) (a) (x a) n n! The remainder of the Taylor Series is lim T n(x) = f(x). n R(x) = f(x) T n (x) (x a) 2 + f (a) (x a) 3 3! K.Maruno (UT-Pan American) Analysis II March 4, 20 5 / 8

16 Taylor s Theorem Theorem If f(x) = T n (x) + R n (x), where T n is the nth-degree Taylor polynomial of f at a and lim n R n(x) = 0 for x a < R, then f is equal to the sum of its Taylor series on the interval x a < R. Taylor s Theorem If f has n+ derivatives in an interval I that contains the number a, then for x in I there is a number z strictly between x and a such that the remainder term in the Taylor series can be expressed as R n (x) = f (n+) (z) (x a)n+ (n + )! K.Maruno (UT-Pan American) Analysis II March 4, 20 6 / 8

17 Remainder of Taylor Series f(x) = e x nth-degree Taylor polynomial of f(x) = e x at 0 The remainder is T n (x) = n f (i) (0) i=0 i! x i = + x + 2! x2 + 3! x3 + + ea n! xn R n (x) = e z (n + )! xn+. If x > 0, then 0 < z < x, so e z < e x. Therefore 0 < R n (x) = e z (n + )! xn+ < e x xn+ (n + )! 0 as n. By the Squeeze theorem, R n (x) 0 as n. K.Maruno (UT-Pan American) Analysis II March 4, 20 7 / 8

18 Remainder of Taylor Series If x < 0, then x < z < 0, so e z < e 0 =. Therefore R n (x) < x n+ (n + )! 0 as n. By the Squeeze theorem, R n (x) 0 as n. Remark:Taylor polynomials of a function f(x) approximate a function f(x). The remainder of Taylor series R n (x) can be used for estimating the error f(x) T n (x) = R n (x). K.Maruno (UT-Pan American) Analysis II March 4, 20 8 / 8