Section 10.7 Taylor series
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1 Section 10.7 Taylor series 1. Common Maclaurin series 2. s and approximations with Taylor polynomials 3. Multiplication and division of power series Math 126 Enhanced 10.7 Taylor Series The University of Kansas 1 / 13
2 Definitions The Taylor series of f (x) at c is the series f (n) (c) T (x) = (x c) n = f (c)+f (c)(x c)+ f (c) (x c) n! 2! n=0 We call the special case where c = 0 the Maclaurin series of f (x) (or just the Taylor series of f (x) at 0). This series is given by T (x) = n=0 f (n) (0) n! x n = f (0) + f (0)(x) + f (0) (x) ! Math 126 Enhanced 10.7 Taylor Series The University of Kansas 2 / 13
3 Definitions The Taylor series of f (x) at c is the series f (n) (c) T (x) = (x c) n = f (c)+f (c)(x c)+ f (c) (x c) n! 2! n=0 We call the special case where c = 0 the Maclaurin series of f (x) (or just the Taylor series of f (x) at 0). This series is given by T (x) = n=0 f (n) (0) n! x n = f (0) + f (0)(x) + f (0) (x) ! Example: The Maclaurin series for f (x) = 1 1 x is n=0 x n with RoC=1. Math 126 Enhanced 10.7 Taylor Series The University of Kansas 2 / 13
4 Find the Maclaurin series for: 1. f (x) = 1 1 x Math 126 Enhanced 10.7 Taylor Series The University of Kansas 3 / 13
5 Find the Maclaurin series for: 1. f (x) = 1 1 x 2. g(x) = e x Math 126 Enhanced 10.7 Taylor Series The University of Kansas 3 / 13
6 We will frequently use: Function Maclaurin Series Radius 1 x n 1 1 x e x sin(x) cos(x) arctan(x) ln(1 + x) n=0 n=0 n=0 x n n! ( 1) n x 2n+1 (2n + 1)! n=0 ( 1) n x 2n (2n)! ( 1) n x 2n+1 2n + 1 ( 1) n x n+1 n=0 n=0 n + 1 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 4 /
7 iclicker Question 1 Determine f (4) ( 1) if the Taylor series for f is T (x) = 3(x + 1) + (x + 1) 2 4(x + 1) 3 + 2(x + 1) (A) (B) 2 (C) (D) 24 (E) 48 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 5 / 13
8 Find the Maclaurin series for the given functions. Remember to include the Radius of Convergence. 1. g(x) = x 2 e x 2 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 6 / 13
9 Find the Maclaurin series for the given functions. Remember to include the Radius of Convergence. 1. g(x) = x 2 e x 2 2. h(x) = x 2 sin(πx) Math 126 Enhanced 10.7 Taylor Series The University of Kansas 6 / 13
10 Find the Maclaurin series for the given functions. Remember to include the Radius of Convergence. 1. g(x) = x 2 e x 2 2. h(x) = x 2 sin(πx) Find the Taylor series about c = 2 for the given functions. Remember to include the Radius of Convergence. 1. f (x) = x 3 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 6 / 13
11 Find the Maclaurin series for the given functions. Remember to include the Radius of Convergence. 1. g(x) = x 2 e x 2 2. h(x) = x 2 sin(πx) Find the Taylor series about c = 2 for the given functions. Remember to include the Radius of Convergence. 1. f (x) = x 3 2. m(x) = ln(x) Math 126 Enhanced 10.7 Taylor Series The University of Kansas 6 / 13
12 iclicker Question 2 Find the coefficients c n of the Taylor series for f (x) = x about a = 4. (A) c n = ( 1) n (2n 1) 2 2n 1 n! (B) c n = ( 1) n (2n 3) 2 2n 1 (C) c n = ( 1) n (2n 3) 2 3n 1 n! (D) c n = (2n 3) 2 3n 1 n! Math 126 Enhanced 10.7 Taylor Series The University of Kansas 7 / 13
13 type: finding values of sums Find the value of the sum for each of the series. ( 1) n 9 n π 2n n! n=1 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 8 / 13
14 type: finding values of sums Find the value of the sum for each of the series. ( 1) n 9 n π 2n n! 2. n=1 ln(2) 4 + ln(2) ln(2) ln(2) Math 126 Enhanced 10.7 Taylor Series The University of Kansas 8 / 13
15 type: finding values of sums Find the value of the sum for each of the series. ( 1) n 9 n π 2n n! n=1 ln(2) 4 n=0 + ln(2) ln(2) ln(2) ( 1) n+1 π 2n+3 4 2n+1 (2n + 1) Math 126 Enhanced 10.7 Taylor Series The University of Kansas 8 / 13
16 type: evaluating limits Evaluate the limits using Taylor series. cos(x) 1 + x lim x 0 x 4 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 9 / 13
17 type: evaluating limits Evaluate the limits using Taylor series. cos(x) 1 + x lim x 0 x 4 2. lim x 0 e x e x x Math 126 Enhanced 10.7 Taylor Series The University of Kansas 9 / 13
18 type: evaluating limits Evaluate the limits using Taylor series. cos(x) 1 + x lim x 0 x 4 2. lim x 0 e x e x x 3. lim y 0 y arctan(y) y 3 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 9 / 13
19 type: evaluating integrals Evaluate the integrals. e x 2 1. x dx 2 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 10 / 13
20 type: evaluating integrals Evaluate the integrals. e x 2 1. dx 2. x x 1 x 8 dx Math 126 Enhanced 10.7 Taylor Series The University of Kansas 10 / 13
21 type: approximations Evaluate 1 0 cos(x 2 ) dx correct to within Math 126 Enhanced 10.7 Taylor Series The University of Kansas 11 / 13
22 type: approximations Evaluate 1 0 cos(x 2 ) dx correct to within Approximate sin(12 ) to 3 decimal places. Math 126 Enhanced 10.7 Taylor Series The University of Kansas 11 / 13
23 iclicker Question 3 Find the sum of the series (A) cos(2) (B) sin( 2) (C) e 2 (D) cos( 2) (E) e 1 2 Math 126 Enhanced 10.7 Taylor Series The University of Kansas 12 / 13
24 miscellanea Homework 3, bonus question #2. Find the first five nonzero terms in the Maclaurin series of the given functions. 1. f (x) = ex 1 x 2. g(x) = ex cos(x) Math 126 Enhanced 10.7 Taylor Series The University of Kansas 13 / 13
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