Taylor Series

Theorem

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

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

Examples I

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

Taylor Series for sinx about 2π

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

Taylor Series for sinx about π

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!-sinx cosx3 (1 / 4!)(x- π/2( 4 1 / 4!1sinx4 0 00cosx5

Taylor Series for sinx about π/2

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

Maclaurin Series for sinx

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

Approximating sin( 2 ○ )

Examples II

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

Maclaurin Series for e x

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

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

Taylor Series for lnx about x = 1

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