Brief Study Guide for Chapter 12 - B   (Sections 12.8 -12.12)

Preview For Chapter 12-B

Power Series

As useful as sequences and series are, the real "power" in series comes when we introduce a variable  x  into the series.  Up to this point in time, the only variable in our series has been the index  n  running from 1 to infinity.  In power series, we keep the index  n  as before and introduce powers of  x,  i.e., xn, into the nth term of the series, hence the name power series.

Many popular functions like trigonometric functions and rational functions have power series representations.  A polynomial and its power series representation are one and the same.  Some functions like the Bessel functions are defined in terms of its power series representations.

The famous Maclaurin series and Taylor series, special cases of power series, have wide applications in physics and engineering.

Calculus 2     Lesson 12.8     Power Series
The following topics are the most important. Typical exercises from Calculus, 5th edition, by James Stewart, are assigned at the end of each objective.  
1. Determine the radius of convergence and interval of convergence for a power series. P789#3,7,11,17,21
Maple Assignment: See Maple Assignment

Calculus 2     Lesson 12.9     Representation of Functions as Power Series
1. Form new power series from existing power series using the following operations on power series and tell the radius or convergence of the new series.
1a. replace (x-a) by b( cx d)k in a series p795#3,5,7
1b. multiply or divide a series by b(x-a)k p795#17,23
1c. add or subtract two series p795#11
1d. differentiate a series p795#13
1e. integrate a series p795#19,27,29
Maple Assignment:  none

Calculus 2    Lesson 12.10    Taylor and Maclaurin Series
1. Create the Taylor or Maclaurin series for a given function. P806#13,15,33
2. From the Taylor or Maclaurin series for a function, form the series of a related function using the operations of section 12.9 above. P807#39,41,43
3. Approximate values of the function using the Taylor or Maclaurin polynomial. P807#37
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 12.11     Binomial Theorem
1. Find the power series for a function that can be written as a binomial, f(x) = (1x)k , where k is any real number 
and |x| < 1.  P811#1,3,7,11
Maple Assignment:  none

Calculus 2     Lesson 12.12     Applications of Taylor Series
1. Find the Taylor or Maclaurin polynomial Tn(x) for a function f at a number a.
Show how well the Tn's approximate f by doing the following.
1a. Graph f and several Tn's on the same set of axes. P819#1,3,5
1b. Evaluate f and several Tn's at the same x value. P819#23
2. For a given function f,
2a. Approximate values of the function using the Taylor or Maclaurin polynomial.
2b. Find an upper bound for the error above by bounding the remainder term using Taylor's inequality. P819#13,15
Maple Assignment:  none

Maple worksheets to be submitted for a grade for Chapter 12 - B 
(Sect 12.8 -12.12)

See Maple Assignment