Mathematics 547, 3 credits, Introductory Analysis I

AU, WI

Prerequisite: Mathematics 345 or equivalent

Catalog Description:

547, 548, and 549 is an integrated sequence in advanced calculus covering
sequences, limits, continuous functions, differentiation, Riemann integral;
infinite series, sequences and series of functions, Taylor series, and
improper integrals.

Purpose of the Course:

547, 548, and 549 is a sequence designed to develop analytic intuition and
proof skills. Student participation is emphasized. One of the primary
purposes of 547 is that the student gain experience with concrete estimates
and inequalities.

Follow-up Course: Math 548

Text: Introduction to Real Analysis, 3rd ed., Bartle/Sherbert
(http://jws-edcv.wiley.com/college/tlp/0,9842,MATHC-MTC-MTX7C-MT70C_0471321486_BKS,00.html)

Topics:

1. Monotone functions. Monotone sequences.

2. Boundedness. Estimates.

3. Definition of the limit of a sequence. Limit rules. Standard examples.

4. Principle of nested intervals. The Bolzano-Weierstrass Theorem. The Cauchy
Criterion. Supremum and infimum.

5. Infinite series. Comparison tests. Ratio and root tests. Integral test.
Absolute convergence.

___________________________________________________________________________

Mathematics 548, 3 credits, Introductory Analysis II

WI, SP

Prerequisite: Mathematics 547

Catalog Description: Continuation of 547

Purpose of the Course:

547, 548, 549 is a sequence designed to develop analytic intuition and proof
skills. Student participation is emphasized.

Follow-up Courses: Math 549 or 551 or 552

Text: Introduction to Real Analysis, 3rd ed., Bartle/Sherbert
(http://jws-edcv.wiley.com/college/tlp/0,9842,MATHC-MTC-MTX7C-MT70C_0471321486_BKS,00.html)

Topics:

1. Conditionally convergent series. Alternating series. Rearrangements.

2. Power series.

3. Continuous functions.

4. Limits of functions.

5. Uniform continuity.

6. Definition of the derivative. Differentiation rules.

7. Mean-Value Theorem.

8. L'Hospital's Rules.

9. Convexity.
___________________________________________________________________________

Mathematics 549, 3 credits, Introductory Analysis III

AU, SP

Prerequisite: Mathematics 548

Catalog Description:

Continuation of 548; the Riemann-Stieltjes integral; an introduction to the
calculus of several variables.

Purpose of the Course:

547, 548, and 549 is a sequence designed to develop analytic intuition and
proof skills. Student participation is emphasized.

Text: Introduction to Real Analysis, 3rd ed., Bartle/Sherbert
(http://jws-edcv.wiley.com/college/tlp/0,9842,MATHC-MTC-MTX7C-MT70C_0471321486_BKS,00.html)

Topics:

1. Taylor's Theorem.

2. Definition of the Riemann integral. A piecewise continuous function is
Riemann integrable. Properties of the integral.

3. The Fundamental Theorem of Calculus. Integration by parts and change of
variables.

4. Exponential and logarithmic function.

5. Improper integrals.

6. Functional sequences and series.

7. Uniform convergence.

8. Power series and analytic functions.

____________________________________THE END_______________________________