Portland Community College | Portland, Oregon

Upon successful completion of the course, students should be able to:

1. Recognize and define sequences in a variety of forms and describe their properties, including the concepts of convergence and divergence, boundedness, and monotonicity. 

2. Recognize and define series in terms of a sequence of partial sums and describe their properties, including convergence and divergence.

3. Recognize series as harmonic, geometric, telescoping, alternating, or p-series, and demonstrate whether they are absolutely convergent, conditionally convergent, or divergent, and find their sum if applicable.

4. Choose and apply the divergence, integral, comparison, limit comparison, alternating series, and ratio tests to determine the convergence or divergence of a series.

5. Determine the radius and interval of convergence of power series, and use Taylor series to represent, differentiate, and integrate functions. 

6. Use techniques and properties of Taylor polynomials to approximate functions and analyze error.

Enjoy a life enriched by exposure to one of humankind's great achievements.

1. Demonstrate an understanding of the concepts of sequences and series and their application to real world problems in:

   • at least two proctored, closed-book, no-notes exams worth at least 50% of the overall grade, one of which is a comprehensive final that is worth at least 25% of the overall grade, consisting primarily of free response questions (although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate)

   • and at least one of the following:

     • Take-home examinations

     • Graded homework problems

     • Quizzes

2. Consistently demonstrate proper notation, documentation, and use of language throughout all assessments and assignments. For proper documentation standards see Addendum.

3. Demonstrate an ability to work and communicate with colleagues, on the topics of sequences and series, in at least two of the following:

   • A team project with a written report and/or in-class presentation

   • Participation in discussions

   • Group activities

• Students will explore numerical methods of approximation.

• Students will learn to determine the convergence status of a given series.

• Students will learn estimation techniques for convergent series.

• Students will learn to model functions with Taylor series and use Taylor Series to solve application problems.

• Classroom activities will include lecture/discussion and group work.

• Students will communicate their results in oral and written form.

• Students will apply concepts to real world problems.

• The use of technology will be demonstrated and encouraged by the instructor where appropriate. Technology will be used (at least) when estimating convergent series.

1. Applications of the Derivative

   The goal is to use numerical methods of approximation.

   1. Use approximation strategies

      1. Formulate linear approximations to estimate f(x) near x = a. 

      2. Apply Newton’s Method to iteratively approximate roots of differentiable functions. 

   2. Use L'Hôpital's Rule to evaluate limits for indeterminate forms including \(\frac{\infty}{\infty}, \frac{0}{0}, \infty - \infty, 0 \cdot \infty, 0^0, 1^\infty, \infty^0\).

2. Sequences and Series

   The goal is to study convergent/divergent sequences and series, and to approximate functions with simpler functions.

   1. Convergent and Divergent Infinite Sequences

      1. Establish convergence and divergence for infinite sequences presented graphically, numerically, and symbolically. 

      2. Use limits to determine the convergence or divergence of an infinite sequence using a variety of strategies, including L’Hôpital’s Rule, the Squeeze Theorem, and the Monotone Convergence Theorem.

   2. Convergent and Divergent Infinite Series

      1. Define conditions for convergent/divergent series. 

      2. Utilize strategies for testing series for convergence/divergence.

         1. Apply the Test for Divergence, Telescoping Series, Integral Test, Comparison Test, Limit Comparison Test, Ratio Test, and Alternating Series Test for various convergent/divergent infinite series.

         2. Estimate the sum of convergent infinite series using partial sums or estimation techniques such as comparing to an integral. 

         3. Estimate the error in using partial sums to approximate the sum of a convergent alternating series. 

      3. Determine sums of convergent series when possible (e.g. geometric series and telescoping series). 

   3. Power Series

      1. Create a new series by using substitution (e.g. by comparing \(\frac{1}{1 + 2x}\) to the series representation of \(\frac{1}{1 - x}\)). 

      2. Integrate or differentiate power series. 

      3. Determine a power series representation by using differentiation or integration (e.g. find the series representation of \(\ln(1 + x)\) by using an appropriate integral). 

      4. Determine the interval and radius of convergence for a power series. 

   4. Taylor Series

      1. Construct a Taylor polynomial of degree 𝑛 to approximate a function about a given center.

      2. Construct and utilize particular Taylor and Maclaurin series and their intervals of convergence, including \(e^{x}, \cos(x), \sin(x)\).

      3. Calculate a bound on the error in using Taylor polynomials to approximate a function by using Taylor’s Inequality (or the Alternating Series Estimation Theorem where appropriate).

3. Optional Topics
   1. The following topics may be of interest to the instructor or the student and can be covered as time allows. 

      1. Binomial Series

      2. Fourier series

      3. The root test

      4. Proof by Induction

All work in this course will be evaluated for your ability to meet the following writing objectives as well as for mathematical content.

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Remember to line up your equal signs.
13. If work is word-processed all mathematical symbols must be generated with a math equation editor.

Emphasis should be placed on using technology such as Desmos and GeoGebra appropriately; such as when computing approximations, graphing curves, or visualizing or checking answers.  Technology should not be used as a substitute for meeting the outcomes and skills for the course that are expected to be done by hand.