Some facts for Final Exam
Preface
The FINAL EXAM will consist of two parts. The first part will cover
the material from Chapter 4 and will be 25-35% of the exam. The second
part will cover some of the important concepts from Chapters 1-3.
Basic Knowledge Base for Part I of the Final
- Convergence of Power Series; radius of convergence, how to deal
with the endpoints of the interval of convergence, the type of
convergence (absolute!, uniform on closed intervals) inside the radius
of convergence
- Convergence of a sequence of functions: definition of pointwise
convergence, definition of uniform convergence.
- Theorem that the uniform
limit of a sequence of continuous functions is continuous.
- Theorem that a limit can be taken inside the integral for
uniform convergence.
- Definition of a uniformly Cauchy sequence of functions
- Theorem that a uniformly Cauchy sequence of functions is guaranteed
to converge uniformly to a function.
- Definition of uniformly convergent series of functions.
- The Weierstrass M-test for the uniform convergence of series of
functions.
- The Theorems that the term-by-term integration or the term-by-term
differentiation of power series gives power series with the same
radius of convergence and which are the integral or derivative
(respectfully) of the function represented by the original series.
As usual all theorems and definitions should be known by you well enough to
use in other problems. The best course of study for this part is to
carefully think through the homework and problems and
really understand it.
Basic Knowledge Base for Part II of the Final
Here is a short version giving some things more emphasis.
- Technique of Mathematical induction
- The Completeness Axiom
- Definition of Inf S and Sup S. Be able to determine the inf
and sup for concrete examples.
- Definition of convergence to a limit of a sequence
(both as a finite number and as plus or minus infinity)
- Definition of a bounded (above, below or both) set and sequence
- The Archimedean principle
- Definition of the limsup(sn) and
liminf(sn)
- Definition of a Cauchy Sequence
- Definition of convergence of an infinite series
- Cauchy Criterion for infinite series
- Definition of absolute convergence of an infinite series
- Convergence Theorem for Bounded Monotone Sequences
- Theorem about the equivalence of a "convergent sequence" and a
"Cauchy sequence".
- The Bolzano-Weierstrass Theorem
- The Comparison, Ratio, Root, Integral and Alternating Series
Tests
- Definition of Uniform Continuity. Be able to use the definitions
of continuity, uniform continuity
- Theorem about the existence of a max and min for a real valued
function which is continuous on a closed and bounded set (interval).
- Intermediate Value Theorem
- Theorem about Continuity on a closed and bounded interval (set)
implying uniform continuity.
- Preservation of Cauchy sequences by uniformly continuous
functions
- Be able to determine the continuity and uniform continuity for concrete functions
Those items comprise the basic knowledge set. Beyond knowing
them, you need to make the connections to the problems at
hand (the hard part). Study the examples and homework and
think about how the connections have been made.