Tests and Solutions

3. **Test
3** - **Test 3 Solutions**

Final Exam Responsibilities

Self-Scheduled – May 11-15

1. **Limit of a Sequence**: Know how to use the limit “laws” and
theorems in section 1, determine the limit of a sequence and when a sequence
diverges – Know how to use the Monotone (Bounded) Sequence (Convergence)
Theorem, p 709 – Practice problems: **Ch 11 Rev** **TRUE-FALSE** p
786:
1,3,11,16
p 787: 1,2,3,6,9; **11.2**: 37

2. **Types
of Infinite Series**: Know how to identify a geometric series, determine
when a geometric
series diverges, and use the formula for its sum when it converges,
how to use the Divergence Theorem p 718, and how to identify and classify (according
to convergence or divergence) a p-series (p 725) – Know how to use algebra
to reduce an infinite series to infinite series involving geometric series – Practice
Problems: **Ch 11 Rev** p 786-787: 1,12,27; **11.2**:
16,22,27,35; **11.3**: 10,29

3. **Compare
Series**: Know how to apply the procedure if you know both the series
for which you
want to determine convergence and a “known” series
with which to compare – know how to choose a series with which to compare
a given series – Practice Problems: **11.4**: 10,11,14,16,19
(Compare the series in problem 11 to the Harmonic series, and
compare the series in
problem 19 to a geometric series whose common ratio is 2/3.)

4. **Alternating
Series Test**: Know how to identify an alternating series, how
to determine whether the Alternating Series Test applies, and how to apply
that test – know how to determine conditional convergence and absolute
convergence and know how to estimate the sum of a convergent series to a
desired degree of
accuracy – Practice problems: **11.5**: p 740: 11,13,14,27

5. **Ratio
and Root Tests**: Know how to apply each of the two tests to given
series of numbers and to power series – Practice
Problems: **11.6**: pp 745-746: 9,10,15,25

6. **Direct
Substitution Method of Integration**: Know how to apply the procedure
if you know the required substitution – Know
how to make reasoned guesses for a substitution and apply the procedure for
your substitution – Practice Problems:
**5.5**: p 420-421: 2,34,36,43,57,62

7.** Integration
by Parts Method of Integration**: Know how to apply the procedure
if you know the required identification of u and dv – Practice
Problems: **7.1**: p 480: 2,21,23,35

8. **Folding
1/7 Sequence**: Know how to apply the 3-step folding procedure and
represent the
result as a sequence of estimates for 1/7 starting
with an
arbitrary guess – Practice
Problem: Start with an arbitrary estimate a[1] and show that the
sequence of estimates generated by repeating the 3-step process
on the result of each preceding 3-step process
converges to 1/7. See
the **diagram** for
the process and **Test 2 solutions**.

9. **Improper
Integrals of Type I**: know how to evaluate an improper integral
of Type I as the limit of a definite integral – Practice
Problems: **7.8**:
20,24

10. **Integral
Test of Convergence of an Infinite Series**: Know how to apply the
Integral test and draw conclusions about the convergence of a related infinite
series
- Practice Problems: **Ch 11 Rev** p 787: 15

11. **Center
of Mass**: Know how to evaluate moments and centers of mass by applying
integrals and symmetry - Practice
Problems: **Ch
8 Rev** p 583: 12

12. **Probability
Density Functions**: Know how to use information about probability
density and a mean to determine probabilities (and information about probability
to determine the mean) - Practice
Problems: **Ch
8 Rev** p 583: 21

13. **Power
Series**: Know how todetermine the interval of convergence of a power
series and how to use algebra, differentiation, and integration to determine
power series representations for functions based on the representations of
"well-known" functions (the functions listed at the bottom of page 767) -
Practice Problems: **11.10** p
770:
26,27,28; **11.9** p 759: 9,14 (14a was done in class)

14. **Taylor
Series**: Know how to generate the Taylor Series for a function using
the Taylor formulas, pp 761-761; know how to evaluate
an integral by replacing one representation of a function by its Taylor series
representation - Practice Problems: **Ch 11 Rev** p 788:
5; **11.9** p 759: 27

15. **Differential
Equations**: Be able to recognize a differential equation and to confirm
that a proposed solution is valid, know how to apply the method of separation
of variables to solve a differential equation,
including ones with specified initial conditions, know how to apply the power
series method for solving a differential equation - Practice problems: **9.3** p
607: 14,19; See
examples 14-17 below for solutions of initial value problems by using power
series.

Examples

1. Calculating the limit
of a sequence whose nth term is in the form of a **Rational
Function**

2. Divergence of the **Harmonic
Series**

3. Balancing Cards (and
boxes) **Hints**

4. A few **Series
Practice
Problems**

5. **Factorial
Exercises** & **Factorial
Solutions**

9. **Integral
Practice Spring 07**

12. Probability and Waiting: **Left
Bank** - **Holden
Call**

17. Calculating
the Taylor
Series (MacLaurin Series) by using the **Taylor
Formulas**

14. Solving an** Initial
Value Problem**, y' + y = 2 cos(t); y(0) = 1, with Power Series

15. Solving **Another** Initial
Value Problem, y'' + y = x^2 + 2; y(0)=0, y'(0)=1, with
Power Series

16. **Yet
Another** Power Series
solution: Now for the IVP: (1+x^2)y'=xy; y(0)=1.

17. **Final
(!) Example** of
using a Power Series to Solve an IVP

Using the
software Package **Maple**

1. * Maple*:
Working with sequences

2. * Maple*:
Working
with infinite series

3. **Maple**: Finding
integrals

4. * Maple*:
Learning to use techniques of integration (Nov
2, 2006, class)

5. **Maple**: Power
Series Radius of Convergence

6. **Maple**: Taylor
Series & Taylor Polynoms.pdf

7. **Maple**: Fourier
Series