Taylor and Maclaurin Series Lesson 9.10

Convergent Power Series Form Consider representing f(x) by a power series For all x in open interval I Containing c Then

Taylor Series If a function f(x) has derivatives of all orders at x = c, then the series is called the Taylor series for f(x) at c. If c = 0, the series is the Maclaurin series for f.

Taylor Series This is an extension of the Taylor polynomials from section 9.7 We said for f(x) = sin x, Taylor Polynomial of degree 7

Guidelines for Finding Taylor Series 1.Differentiate f(x) several times Evaluate each derivative at c 2.Use the sequence to form the Taylor coefficients Determine the interval of convergence 3.Within this interval of convergence, determine whether or not the series converges to f(x)

Series for Composite Function What about when f(x) = cos(x 2 )? Note the series for cos x Now substitute x 2 in for the x's

Binomial Series Consider the function This produces the binomial series We seek a Maclaurin series for this function Generate the successive derivatives Determine Now create the series using the pattern

Binomial Series We note that Thus Ratio Test tells us radius of convergence R = 1 Series converges to some function in interval -1 < x < 1

Combining Power Series Consider We know So we could multiply and collect like terms

Assignment Lesson 9.10 Page – 29 odd