As indicated by the course number, this is one of our transitional courses. The course goal is to gently move students from the computational orientation they encountered in calculus to the more rigorous approach they will need in 4000 level courses. The key word here is gently. Students at this level are not ready for a pure definition-theorem-proof approach. Rather, they should gradually be convinced of the value of precise definitions and careful notation, and constantly asked to construct their own concrete examples and counterexamples. By the end of the course, students should be able to write epsilon-N limit proofs, organize the estimates necessary in applying Taylor's Theorem, and have some feeling for the meaning of uniform convergence.
The prerequisite for the course is MATH 2260, Calculus for students
in science and engineering. The current 2260 syllabus does
include treatment of some sequence and series topics, e.g., application
of various series convergence tests. You should not skip such
topics in 3100, but rather take advantage of students' prior exposure
to help them understand why the techniques they already know actually
work.
Students can take 3000 (linear algebra), 3100, and 3200 (introduction to higher mathematics) in any order. This means you cannot assume prior exposure to induction or other proof writing techniques at the beginning of the course. (In the past, students taking 3000 and/or 3200 before 3100 have done better overall, but that may have changed now that 2260 covers some material on sequences and series.)
Just before students start meeting with their academic advisers to
register for the following term, you should spend some time in class
mentioning possible sequels to 3100. See for example the flow
chart at http://www.math.uga.edu/%7Ecurr/prereqs.html.
Course notes for MATH 3100, available online and suitable for use as
texts:
| Topic |
Comments |
Adams Sections |
Azoff Sections |
Weeks |
| Ordered fields |
Axiomatic treatment is optional, but developing
computational facility with inequalities and absolute values is a must |
1.1prob12, 1.4 |
1.1-1.3 |
1.5 |
| Completeness and the reals |
This can be based on least upper bounds (Azoff) or on
monotone sequences (Adams) |
1.6 |
1.4,1.6 |
|
| Definition of sequence, examples, recursion and induction |
Newton's method provides a good example of recursion. |
1.1,1.2,1,3 |
2.1,1.5 |
1 |
| Definition of sequence convergence (epsilon-N) |
Do not rush this. |
1.4 |
2.2 |
1 |
| Algebra of sequential limits |
Some of these results can be stated without proof. |
1.5 |
2.3 |
1/2 |
| Monotone sequences |
Regardless of whether convergence of bounded monotone
sequences is taken as an axiom (Adams) or based on least upper bounds
(Azoff), it is important for students to have an intuitive feeling for
this. |
1.6 |
2.4 |
1/2 |
| Subsequences |
Use honest function notation to emphasize the fact that a
subsequence is a composite of functions. |
1.3 |
2.5 |
1/2 |
| Cauchy sequences |
This concept (and the preceding one) can be applied in a
negative way to establish divergence of sequences. |
1.6 |
2.6 |
1/2 |
| Applications of l'Hopital's rule |
Use of calculus results makes this course less
self-contained, but the computational benefits are well-worth the
sacrifice. |
1.5 |
2.7 |
1/2 |
| Applications to calculus | This topic can be postponed till
later in the course if desired. The idea is to expose students to
J. Hollingsworth's "three
C's": continuity, connectedness (the intermediate value theorem), and
compactness (the maximum value theorem) |
1.7 |
Chap 4 |
1 |
| Definition of series convergence |
The distinction between sequences of terms and sequences of
partial sums needs to be emphasized repeatedly. |
2.1 |
3.1 |
1 |
| Geometric and telescoping series |
Use these to illustrate the preceding definition. |
2.1 |
3.1 |
|
| Series convergence tests (n'th term, comparison,
limit-comparison, integral, root, ratio) |
Students should have some prior experience with the mechanics
of these tests, so the emphasis should be on why they work. |
2.2 |
3.2, 3.4,3.5 |
1.5 |
| Absolute and conditional convergence |
Include an informal discussion of series rearrangement. |
2.3 |
3.6 |
1/2 |
| Mixed convergence problems |
Hopefully, this can be covered relatively quickly because of
MATH 2260 |
2.3 |
3.7 |
1/2 |
| Taylor's theorem (theory) |
While you will probably want to include a proof, it is most
important for students to understand what the Theorem says. |
3.2 |
5.1 |
1/2 |
| Taylor's theorem (computations and estimates) |
Move from simple (find specified Taylor polynomials) to
intermediate (estimate errors of specified approximations) to
"advanced" (decide how many terms are necessary for given accuracy). |
3.2 |
5.2 |
1 |
| Power series: domains of convergence |
Do not emphasize formulas for radii of convergence, but have
students apply convergence tests for numerical series. |
2.4 |
6.1 |
1/2 |
| Uniform convergence and continuity | This should be regarded as a capstone for the course. |
3.1 |
6.2 |
1 |
| Operations on power series and applications to Taylor series |
You may go easy on these proofs, but
be sure students can get Taylor series for functions like arctan(x^3)
from
the series for 1/(1+x). |
3.2 |
5.3 |
1/2 |
| Optional |
Decimal expansions, complex series and Euler's formula. |
3.3 |
3.3, Ch7 |