Review for Third Hour Exam,  Calculus II

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The following is a list of some questions on some of the topics we've
covered in this course recently.  Some of this material will also appear
on the exam.  Material not mentioned here may also be covered on the exam.

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1.  What is a sequence?  Give an example.  What does it mean for a sequence to 
converge?  Give several examples of sequences which converges and of those 
which do not.  

2.  What is an indeterminate form?  State L'Hopital's Rule.  How can you use 
L'Hopital's Rule to determine the convergence of sequence?  Give several 
examples of the use of L'Hopital's Rule.  

3.  What is a series?  Give an example.  What is a partial sum of a series?  
What is the sequence of partial sums of a series?  What does it mean for a 
series to converge?  Give several examples of series which converge and those 
which do not.  Carefully explain the difference between the sequence of terms 
of the series, the series itself, the sequence of partial sums of the series, 
and the limit of the series.

4.  What is a geometric series, a p-series, an alternating series?  Give 
examples of each.  When does each of these series converge?

5.  What does it mean for a series to converge absolutely, to convergence 
conditionally?  Give an example of each type of series.

6.  State each of the convergence tests: Divergence test, Integral test, 
Comparison test, Limit comparison test, Ratio test, Absolute convergence test, 
Alternating series test.  Be sure to state when the test applies and what 
conclusion you can and cannot draw from the test.  Give an example of the use 
of the test.

7.  What is the error formulas for an alternating series?  Give examples 
showing how to estimate the sum of an alternating series by using an error term.

8.  What is a Taylor polynomial?  What are the criteria used to derive the 
Taylor polynomial of degree k about the point a?  Give an example.  What is the 
Taylor remainder formula for this polynomial?  Give an example.

9.  What is a power series?  Give an example.  How can they be used?  What is 
the radius of convergence of a power series?  How do you compute it?  Give 
several examples.

Essay: Write a short essay on the Taylor polynomials and the Taylor series of a 
function f(x).  Explain what they are, how they are computed, and the 
relationship between them; include a description of the radius of convergence 
of the Taylor series; describe the Taylor remainder formula and its 
relationship to the covergence of the Taylor series to f(x).