Exam Review

Exam 3 Review

Math 208, Fall 2008

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This version: Monday, October 20, 2008, 9:34 AM

Exam 3 will be given in class on (changed date) Monday, November 24. It will take all of the class that day.

You will do the exam on your own paper. Bring plenty of blank paper.
Missing the exam will give you a grade of zero.
DO NOT miss the exam for any reason.
The exam will cover the material covered in class through (changed date) Monday, November 17: Section 11.4 through Section 12.6. However, you will not be tested on Newton's Method or Section 11.5.

Appropriate end-of-chapter review problems include pages 641–643 # 1–24, 33–38, 40–46, 49–70, 75–84, 120.

You will definitely have problems where you must explain why various series converge or diverge. On such problems you are expected to cite (that is, give by name) the convergence tests as you apply them. These are basically proofs, and clear reasoning is required. More details are below. If you are in any doubt as to what I expect, you should see me in my office hours.

You might have to prove the limit of a sequence using an ε-N proof. Also, you should be able to give the precise definition for the limit of a sequence. You may use either the version with n ≥ N (as in your book) or n > N (as in class—this one is easier).

Know the definition of the limit of a sequence (very similar to the limit of a function as x goes to ∞). See above note.
Be able to compute various limits of sequences. Be familiar with the Comparison Test for sequences (page 543); note that it's quite different from our other comparison tests.

One consequence of this Comparison Test is that, as i goes to ∞, a_i goes to zero if and only if |a_i| goes to zero. This can be useful when applying the i^th term test, the Alternating Series Test, etc.

There will be NO questions on Exam 3 (or on the Final Exam) covering estimating of zeros of functions (Newton's Method) or estimating the value of an integral (Riemann sums, the Trapezoid Method, Simpson's Method).

Be familiar with how series are defined, i.e. as the limit of the sequence of partial sums. A series converges if this limit exists, and otherwise we say it diverges.
Be able to recognize and find the sum of geometric series.
Sums and constant multiples are easy. (See top of page 566.) There is no such rule for products or quotients, though!
Be able to classify every series Σ a_i as one of these:

Σ a_i converges absolutely. That means Σ a_i converges and also Σ |a_i| converges. (Note that if Σ |a_i| converges, it is automatic that Σ a_i converges, by the Direct Comparison Test.)
Σ a_i converges conditionally. That means Σ a_i converges but Σ |a_i| diverges.
Σ a_i diverges. (In this case, automatically Σ |a_i| diverges by the Direct Comparison Test.)

Be sure you are VERY familiar with the tests we have studied in class: See Convergence Tests and Convergence Test Summary. (These are PDF Files.)
Be able to show how you use these tests by explaining why and how they apply. When proving convergence or divergence, you must cite the test (i.e. give it by name).
Don't mix up the Ratio Comparison Test with the Ratio/Root Tests.

The Ratio Comparison Test works with two series, showing they both converge or both diverge if the limit of a_i/b_i is positive and not ∞ (whether bigger or smaller than 1!).
The Ratio/Root Tests are used with a single series only, and you look to see if the limit is less than or greater than 1.

Note that we sometimes refer to the Direct Comparison Test as the Comparison Test, and it's quite similar to the Comparison Test for improper integrals (but quite different from the one on sequences mentioned above!).
Be familiar with factorial notation (often when using Ratio Test):

Power series: the power series Σ a_i(x – x₀)ⁱ centered around x₀ is considered to be a function of the variable x (with constants a_i and x₀).

Finding the radius of convergence R: usually the Ratio Test (occasionally the Root Test) shows convergence or divergence at most points. This generally tells you the radius of convergence R.
If R is the radius of convergence, then the power series converges absolutely at all x satisfying |x – x₀| < R and diverges at all x satisfying |x – x₀| > R
Finding the exact domain will generally require looking at the two points satisfying |x – x₀| = R (i.e. x = x₀ + R and x = x₀ – R) individually. Many tests might apply, but the Ratio and Root Tests never help here (because they are "crude" tools).
Manipulation of power series

Addition, subtraction, or multiplication of two series; also multiplication of a series by a constant
Differentiation of power series (remember that the constant term has derivative zero, so there's one term fewer)
Integration of power series (remember + C and, if appropriate, evaluate C)
Finding formulas for functions by using the above manipulations on "standard" series listed above

Remember that some problems are easier to solve using "sigma" notation Σ a_i(x – x₀)ⁱ; others are easier by writing out the first many terms a₀ + a₁(x – x₀) + a₂(x – x₀)² + a₃(x – x₀)³ + a₄(x – x₀)⁴ + .... Use whatever notation seems best to you in a given problem.

Taylor series are just specific examples of power series: beginning with some function f, we define the power series Σ a_i(x – x₀)ⁱ by taking a_i = [f⁽ⁱ⁾(x₀) / i!], where f⁽ⁱ⁾ is the i^th derivative (and f⁽⁰⁾ = f)
The Maclaurin series is the Taylor series centered around x₀ = 0.
Memorize the Maclaurin series for 1/(1 – x), e^x, sin x, cos x, and be sure to know the correct domain for each one.
In Section 12.5, you can ignore the binomial series and its applications, since we skipped this material.

As you should already know, i is the square root of –1, i.e. i² = –1.
The Euler formula: for all complex numbers z, e^iz = cos z + i sin z. The exponential, sine, and cosine functions are defined by power series, but correspond to the ordinary functions for real numbers.
Be able to express a complex number in the form a + bi ("rectangular form") and also in the form reⁱ^θ ("polar form"). The Euler formula and/or a graph allow you to make conversions.

Revised Monday, October 20, 2008. E-mail corrections, suggestions to mmaltenfort@ccc.edu