Taylor and Maclaurin Series Convergent Power Series Form

Taylor and Maclaurin Series

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