Course Outline for Series Calculus and Linear Algebra

Prerequisite Course (s):
A grade of "C" or better in MTH 252 or equivalent

Required Text(s):
Stewart, Calculus: Concepts and Contexts, 3^rd Edition, 2005, Thomson
TenEyck and Ferguson, Linear Algebra Packet

Other Requirements:
Graphing calculator. TI-86, TI-89, or TI-92 recommended

Course Description:
Combines topics from linear algebra and infinite series. Includes geometric, Taylor and
Fourier Series work with applications; and systems applications using matrices and
determinants.

Performance Based Learner Outcomes:
Upon successful completion of the course, students should be able to:

1. Create mathematical models of abstract and real world situations using geometric,
power , Taylor and Fourier series, and situations involving linear systems using matrices
and determinants.
2. Use inductive reasoning to develop mathematical conjectures involving infinite series
models and linear systems modeled with matrices. Use deductive reasoning to verify
and apply mathematical arguments involving these models.
3. Use mathematical problem solving techniques involving infinite series and linear
systems using matrices.
4. Make mathematical connections and solve problems from other disciplines involving
infinite series and linear systems using matrices.
5. Use oral and written skills to individually and collaboratively communicate about
infinite series and their behavior, and about linear systems using matrices and
determinants.
6. Use appropriate technology to enhance mathematical thinking and understanding, to
solve mathematical problems involving infinite series models and to solve problems
involving linear systems using matrices and determinants.
7. Do project that encourage independent, nontrivial exploration of infinite series and
linear systems application and models.

Course Content:

I. Number Sense
A. Determine subsequent values of a given sequence according to apparent pattern
B. Write symbolic model (general nth term) of series given numeric data
C. Predict convergence of series given numeric data
D. Evaluate determinants of matrices

II. Symbolism
A. Given the general term of sequence or series, write numeric list or sum of series
B. Evaluate sums of series symbolically
C. Test for convergence of series symbolically
D. Write the Taylor series approximations for given functions
E. Given symbolic form of a square wave , write its Fourier series approximation
F. Use Taylor series approximations of functions in differentiation , integration, and
evaluation of limits of function
G. Apply the algebra of matrices , including scalar and matrix multiplication , matrix
addition, transposition, evaluation of determinants and determination of eigenvalues
and eigenvectors

III. Problem Solving
A. Use graph analysis to predict convergence of series
B. Solve problems modeled by geometric series
C. Use graph analysis to solve linear systems of equations in R ²
D. Use inductive and deductive reasoning to generalize solutions in R² to linear systems
in Rⁿ
E. Create graphical interpretation of solution of linear systems in R³
F. Solve a variety of problems in linear algebra including linear programming,
difference equations and Markov chain applications

IV. Technology
A. Use calculator or CAS programs to sum series
B. Use calculator or CAS to predict convergence of sequences and series
C. Use CAS to evaluate and graph Fourier approximation of a wave
D. Solve equations using appropriate technology
E. Use graphing calculator or CAS to determine the solutions of systems of linear
equations, and interpret and investigate the characteristics of solutions, individually
and in groups.

V. Communication
A. Express interpretations of local and global behavior of a series in written and oral
form
B. Express interpretations of solutions of linear systems in written and oral form
C. Explain the reasoning used to arrive at a mathematical conclusion
D. Read, write, hear and speak mathematical ideas, individually and in groups

Course Content Outline:

I. Infinite Series
A. Sequence and series patterns
B. Tests for Convergence (including n^th term, ratio, comparison, and integral)
C. Geometric Series
D. Binomial Series
E. Taylor Series
F. Fourier Series
G. Applications of infinite series
H. Error analysis

II. Systems of Linear Equations
A. Row Reduction and Echelon Forms
B. Vector Equations
C. Matrix Equations
D. Solution Sets
E. Linear Independence
F. Linear Transformations

III. Matrix Algebra
A. Matrix Operations
B. Inverses and Their Characteristics
C. Subspaces of Rⁿ

IV. Determinants

V. Vector Spaces
A. Dimensions
B. Nullspace
C. Bases
D. Coordinate Systems
E. Rank

VI. Eigenvalues and Eigenvectors
A. Characteristic Equation
B. Diagonalization
C. Linear Transformations