Chap11_Sec10

# Chap11_Sec10 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

This preview shows pages 1–22. Sign up to view the full content.

11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
In section 11.9, we were able to find power series representations for a certain restricted class of functions. INFINITE SEQUENCES AND SERIES
Here, we investigate more general problems. Which functions have power series representations? How can we find such representations? INFINITE SEQUENCES AND SERIES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
11.10 Taylor and Maclaurin Series INFINITE SEQUENCES AND SERIES In this section, we will learn: How to find the Taylor and Maclaurin Series of a function and to multiply and divide a power series.
TAYLOR & MACLAURIN SERIES We start by supposing that f is any function that can be represented by a power series 2 0 1 2 3 4 3 4 ( ) ( ) ( ) ( ) ( ) ... | | f x c c x a c x a c x a c x a x a R = + - + - + - + - + - < Equation 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES Let’s try to determine what the coefficients c n must be in terms of f . To begin, notice that, if we put x = a in Equation 1, then all terms after the first one are 0 and we get: f ( a ) = c 0
TAYLOR & MACLAURIN SERIES By Theorem 2 in Section 11.9, we can differentiate the series in Equation 1 term by term: 2 1 2 3 3 4 '( ) 2 ( ) 3 ( ) 4 ( ) ... | | f x c c x a c x a c x a x a R = + - + - + - + - < Equation 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES Substitution of x = a in Equation 2 gives: f’ ( a ) = c 1
TAYLOR & MACLAURIN SERIES Now, we differentiate both sides of Equation 2 and obtain: 2 3 2 4 ''( ) 2 2 3 ( ) 3 4 ( ) ... | | f x c c x a c x a x a R = + × - + × - + - < Equation 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES Again, we put x = a in Equation 3. The result is: f’’ ( a ) = 2 c 2
TAYLOR & MACLAURIN SERIES Let’s apply the procedure one more time.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES Differentiation of the series in Equation 3 gives: 3 4 2 5 '''( ) 2 3 2 3 4 ( ) 3 4 5 ( ) ... | | f x c c x a c x a x a R = × + × × - + × × - - < Equation 4
TAYLOR & MACLAURIN SERIES Then, substitution of x = a in Equation 4 gives: f’’’ ( a ) = 2 · 3 c 3 = 3! c 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES By now, you can see the pattern. If we continue to differentiate and substitute x = a , we obtain: ( ) ( ) 2 3 4 ! n n n f a nc n c = × × ××××× =
TAYLOR & MACLAURIN SERIES Solving the equation for the n th coefficient c n, we get: ( ) ( ) ! n n f a c n =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES The formula remains valid even for n = 0 if we adopt the conventions that 0! = 1 and f (0) = ( f ). Thus, we have proved the following theorem.
TAYLOR & MACLAURIN SERIES If f has a power series representation (expansion) at a , that is, if then its coefficients are given by: Theorem 5 0 ( ) ( ) | | n n n f x c x a x a R = = - - < ( ) ( ) ! n n f a c n =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TAYLOR & MACLAURIN SERIES Substituting this formula for c n back into the series, we see that if f has a power series expansion at a , then it must be of the following form. Equation 6
TAYLOR & MACLAURIN SERIES Equation 6 ( ) 0 2 3 ( ) ( ) ( ) ! '( ) ''( ) ( ) ( ) ( ) 1! 2! '''( ) ( ) 3! n n n f a f x x a n f a f a f a x a x a f a x a = = - = + - + - + - +×××

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document