MTH 265 Introduction to Series and Power Series

Lecture Hours: 2
Credits: 2

Convergence and divergence of numerical series, including geometric series. Series of functions. Power series and their analytic properties. Taylor series expansions and Taylor polynomials.

Prerequisite: Placement into WR 115 or completion of WR 090 and MTH 252 with a grade of C or better; or consent of instructor.
Student Learning Outcomes:

Recognize and apply the basic convergence tests for series, including the comparison test, integral test, alternating series test, ratio test and root test.
Determine the radius and interval of convergence for a power series.
Apply term-by-term differentiation and integration when valid.
Approximate functions using Taylor polynomials or partial sums of infinite series

Content Outline

Complex Numbers
- Arithmetic of Complex Numbers
- Roots of Polynomials and Fundamental Theorem of Algebra
- Conjugation
- Euler’s Identity
Series
- Convergence and Divergence
- Geometric Series
- Series of Functions
Convergence Tests
- Divergence Test
- Integral Test
- Alternating Series Test
- Comparison and Limit Comparison Tests
- Absolute Convergence
- Root and Ratio Tests
Power Series
- Basic Ideas
- Intervals of Convergence
- Analytic Properties of power series
Taylor Series
- Taylor Polynomials
- Approximating Functions