1. Taylor Series

2. Theorem

3. Definition The series is called the Taylor series of f about c (centered at c)

4. Definition The series is called the Maclaurin series of f about c (centered at c) Thus a Maclaurin series is a Taylor series centered at 0

5. Examples I

6. Example (1) Taylor Series for f(x) = sinx about x = 2π a n (x- 2π( n a n =f (n) (2π) / n! f (n ) (2π) f (n) (x)n 000sinx0 (1/1!)(x- 2π( 1 1/1!1cosx1 000-sinx2 (-1 / 3!)(x- 2π( 3 -1 / 3!-cosx3 000sinx4 ) 1/ 5!)(x- 2π( 5 1/ 5!1cosx5

7. Taylor Series for sinx about 2π

8. Example (2) Taylor Series for f(x) = sinx about x = π a n (x- π( n a n =f (n) (π) / n! f (n) ( π )f (n) (x)n 000sinx0 (-1/1!)(x- π( 1 -1/1!cosx1 000-sinx2 (1 / 3!)(x- π( 3 1 / 3!1-cosx3 000sinx4 1-)/ 5!)(x- π( 5 1-/ 5!cosx5

9. Taylor Series for sinx about π

10. Example (3) Taylor Series for f(x) = sinx about x = π/2 a n (x- π/2( n a n =f (n) (π/2) / n! f (n) ( π/2 )f (n) (x)n (x- π/2( 0 =1 1 / 0!1sinx0 0 00cosx1 (-1 / 2!)(x- π/2( 2 -1 / 2!-sinx2 0 00-cosx3 (1 / 4!)(x- π/2( 4 1 / 4!1sinx4 0 00cosx5

11. Taylor Series for sinx about π/2

12. Example (4) Maclaurin Series for f(x) = sinx a n x n a n =f (n) (0) / n! f (n ) (0) f (n) (x)n 000sinx0 (1/1!) x 1/1!1cosx1 000-sinx2 (-1 / 3! ) x 3 -1 / 3!-cosx3 000sinx4 (1 / 5! ) x 5 1/ 5!1cosx5

13. Maclaurin Series for sinx

14. Example(5) The Maclaurin Series for f(x) = x sinx

15. Examples II

16. Example (1) Maclaurin Series for f(x) = e x a n x n a n =f (n) (0) / n! f (n ) (0) f (n) (x)n 11 /0!1exex 0 (1 / 1!) x 1 /1!1exex 1 (1 / 2!) x 2 1 /2!1exex 2 (1 / 3! ) x 3 1 / 3!1exex 3 (1 / 4!) x 4 1 /4!1exex 4 (1 / 5! ) x 5 1/ 5!1exex 5

17. Maclaurin Series for e x

18. Example (2) Find a power series for the function g(x) =

20. Example (3) TaylorSeries for f(x) = lnx about x=1 a n x n a n =f (n) (1) / n! f (n ) (1) f (n) (x) 000 lnx 0 (x -1) 11=0! x -1 1 (-1/2) (x -1) 2 -1!/2!-1! -x -2 2 (1/3 ) (x -1) 3 2!/3!2! (-1)(-2)x -3 3 (-1/4 (x -1) 4 -3!/4!-3! (-1)(-2)(-3)x -4 4 (1/5 ) (x -1) 5 4!/5!4! (-1)(-2)(-3)(-4)x -5 5

21. Taylor Series for lnx about x = 1

22. I-1 Taylor Series for f(x) = cos2x about x = 5π a n (x- 5π( n a n =f (n) (5π) / n! f (n) ( 5π )f (n) (x)n 1/0! (x-5π) 0 =1 1/0!1cos2x0 0 00-2 sin2x1 - 2 2 / 2! (x-5π) 2 - 2 2 / 2!- 2 2 -2 2 cos2x2 0 002 3 sin2x3 2 4 / 4! (x-5π) 4 2 4 / 4!2424 2 4 cos2x4 0 00-2 5 sin2x5

23. Taylor Series for cos2x about 5π

24. I-2 Taylor Series for f(x) = cosx about x = 3π/4 a n (x- 3π/4 ( n a n = f (n) (3π/4) / n! f (n) (3π/4)f (n) (x)n - 1/√2 / 0!(x-3π/4) 0 - 1/√2 / 0! - 1/√2cosx 0 - 1/√2 / 1!(x-3π/4) 1 -1/√2 / 1! - 1/√2- sinx 1 1/√2 / 2!(x-3π/4) 2 1/√2 / 2! -(-1/√2) = 1/√2 - cosx 2 1/√2 / 3!(x-3π/4) 3 1/√2 / 3! 1/√2sinx 3 - 1/√2 / 4!(x-3π/4) 4 - 1/√2 / 4! - 1/√2cosx 4 - 1/√2 / 5!(x-3π/4) 5 - 1/√2 / 5! - 1/√2- sinx 5

25. Taylor Series for cosx about 3π/4

26. I.3 Taylor Series for f(x) = cosx about x = π/6 a n (x- π/6( n a n =f (n) (π/6) / n! f (n) ( π/6 )f (n) (x)n √3/2 / 0!(x-π/6) 0 √3/2 / 0! √3/2cosx 0 - 1/√2 / 1!(x-π/6) 1 - ½ / 1! - 1/2- sinx 1 1/√2 / 2!(x-π/6) 2 - √3/2 / 2! - √3/2- cosx 2 1/√2 / 3!(x-π/6) 3 ½ / 3! 1/2sinx 3 - 1/√2 / 4!(x-π/6) 4 - √3/2 / 4! - √3/2cosx 4 - 1/√2 / 5!(x-π/6) 5 - ½ / 5! - 1/2- sinx 5

27. Taylor Series for cosx about π/6

28. I.4 Taylor Series for f(x) = e 3x about x = 5 a n (x- 5( n a n =f (n) (π) / n! f (n) ( 5 ) f (n) (x) n e 15 / 0! (x-5) 0 e 15 / 0! e 15 e 3x 0 3 e 15 / 1! (x-5) 1 3 e 15 / 1! 3 e 15 3e 3x 1 3 2 e 15 / 2! / 1! (x-5) 2 3 2 e 15 / 2! 3 2 e 15 3 2 e 3x 2 3 3 e 15 / 2! / 1! (x-5) 3 3 3 e 15 / 3! 3 3 e 15 3 3 e 3x 3 3 4 e 15 / 2! / 1! (x-5) 4 3 4 e 15 / 4! 3 4 e 15 3 4 e 3x 4 3 5 e 15 / 2! / 1! (x-5) 5 3 5 e 15 / 5! 3 5 e 15 3 5 e 3x 5

30. Homework