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. |
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Calculus
2 Lesson 12.8
Power Series |
| The following topics
are the most important. Typical exercises from Calculus, 4th
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.
P778#3,7,11,17,21 |
| Maple
Assignment: Bessel.mws |
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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 p783#3,5,7 |
| 1b. |
multiply or divide a series by
b(x-a)k p784#19,27 |
| 1c. |
add or subtract two series p784#11 |
| 1d. |
differentiate a series p784#13 |
| 1e. |
integrate a series p784#17,27,29 |
| Maple
Assignment: none |
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Calculus
2 Lesson 12.10 Taylor and
Maclaurin Series |
| 1. |
Create the Taylor or Maclaurin
series for a given function. P794#13,15,31 |
| 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. P794#37,39,41 |
| 3. |
Approximate values of the function
using the Taylor or Maclaurin polynomial. P794#35 |
| Maple
Assignment: TaylorSe.mws |
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Calculus
2 Lesson 12.11
Binomial Theorem |
| |
Although it is very short (only two
pages), this section is not in the syllabus and thus we omit
it. It is, none the less, a widely used topic and may be
of benefit to know. |
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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. P806#1,3,5 |
| 1b. |
Evaluate f and several Tn's
at the same x value. P807#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.
P807#13,15 |
| Maple
Assignment: none |
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