Taylor and Maclaurin Series

1
Taylor and Maclaurin Series

• Lesson 9.10

2
Convergent Power Series Form

• Consider representing f(x) by a power series
• For all x in open interval I
• Containing c
• Then

3
Taylor Series

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

4
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

5
Guidelines for Finding Taylor Series

• Differentiate f(x) several times
• Evaluate each derivative at c
• Use the sequence to form the Taylor coefficients
• Determine the interval of convergence
• Within this interval of convergence, determine
  whether or not the series converges to f(x)

6
Series for Composite Function

• What about when f(x) cos(x2)?
• Note the series for cos x
• Now substitute x2 in for the x's

7
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

8
Binomial Series

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

9
Combining Power Series

• Consider
• We know
• So we could multiply and collect like terms

10
Assignment

• Lesson 9.10
• Page 685
• 1 29 odd