Some information about Midterm 2.
The test is going to cover Sections from 3.3 to 5.3 in the syllabus File:Syl ma341.pdf.
There will be one multiple choice problem and five problems with parts. On this test, the multiple choice will have definitions AND formulations of important theorems. In most cases, important theorems have a special name in the book, e.g., the Bolzano-Weierstrass Theorem. There will be at least one proof on the exam. I will select the theorem that have a not very hard proof. Below, I will try to outline the most important results to review --- this might will be one of the listed theorems. The rest of the problems will be similar to the homework assignment. Keep in mind that it is quite possible that partial credit will be given for stating definitions correctly. So if you do not know how to approach the problem, you may want to start from reciting the definitions that are relevant to the problem. Usually, this also helps to get an idea about the solution.
I will now try to go over each section and outline important theorems.
3.3 Monotone sequences: definition of monotone sequence. Monotone Convergence Theorem. Algorithm for finding square root. Euler's number.
3.4 Subsequences: Definition of a subsequence. Theorem 3.4.2 and 3.4.4. The Monotone subsequence theorem. The Bolzano-Weierstrass Theorem.
3.5 The Cauchy Criterion: Definition of a Cauchy sequence. Lemma 3.5.3. Cauchy Convergence Criterion. Contractive sequence. Theorem 3.5.8.
3.7 Introduction to infinite series. Definition of an infinite series. Convergence of the series. The n-th term test. Cauchy Criterion for series. Theorem 3.7.5. Harmonic and geometric series. Comparison test.
4.1 Limits of functions. Definition of the limit. Definition of a cluster point. Theorems 4.1.2 and 4.1.5. Sequential criterion for limits.
4.2 Limit theorems. Theorems 4.2.4 and 4.2.6. The Squeeze Theorem.
4.3 Some extensions of limit concept. Definitions of one sided limits, limits at infinity, infinite limits. Limit theorems.
5.1 Continuous functions. Definition of continuous function. sequential criterion for continuity. Functions continuous on a set. Examples of continuous functions. Dirichlet's and Tomae's functions.
5.2 Combinations of Continuous functions. Theorems 5.2.1 and 5.2.2. Compositions of continuous functions. Theorem 5.2.6.
5.3. Functions continuous on intervals: definition of a bounded function. Boundedness theorem. Maximum minimum theorem. Location of roots theorem. Bolzano's intermediate theorem. Theorem 5.3.9.
Good luck on the test!
