Study guide for the midterm
(MATH 6102-001, Spring 2013)
- Definitions and notions to remember:
Lower bounds and upper bounds (Definition 2.2.2), Least Upper Bound Property (Definition 2.2.3), Axiom for the Real Numbers (Axiom 2.2.4), intervals (Definition 2.3.6), absolute value (Definition 2.3.8), generalized infinite decimal fractions (Definition 2.8.4), base p representation of real numbers (Definition 2.8.7), eventually repeating base p representation (Definition 2.8.9), sequence (Definition 8.2.1), convergence of sequences (Definition 8.2.2), bounded sequence (Definition 8.2.5), divergence to infinity (Definition 8.2.14), (strictly) increasing and decreasing sequences (Definition 8.3.1), subsequence (Definition 8.3.5, just be able to recognize them, there is no need to memorize the formal definition), Cauchy sequence (Definition 8.3.10), series (Definition 9.2.1), convergence and divergence of partial sums (Definition 9.2.2).
- Exercises you should be able to perform:
Rewrite a rational number as an eventually repeating continued fraction, compute limits of rational expressions of n, find the limit of rn (Exercise 8.2.13), prove a sequence is monotone (also see the homework exercises).
- Statements you should be able to prove:
Basic properties of inequalities (Lemma 2.3.3), Properties of intervals (Lemma 2.3.7), Basic properties of the absolute value (Lemma 2.3.9), (generalized) triangle inequality (Exercise 2.5.3), everything about geometric sequences (see Exercise 2.5.12 and your notes), least upper bound property is equivalent to the greatest lower bound property (Lemma 2.6.2, Theorem 2.6.4, and your notes!), generalized decimal fractions with integer part zero (Lemma 2.8.3 and parts (1) through (2) of Lemma 2.8.5 ), limit of a sequence is unique (Lemma 8.2.3), a convergent sequence is bounded (Lemma 8.2.6), multiplying a bounded sequence with a sequence that goes to zero yields a sequence that goes to zero (Lemma 8.2.7), operations on convergent sequences (Theorem 8.2.9 + your notes for the reciprocal !) limits and inequalities (Theorem 8.2.11), an increasing sequence that is bounded from above is convergent, increasing unbounded sequence diverges to infinity (Theorem 8.3.3), Monotone Convergence Theorem (Corollary 8.3.4), every sequence has a monotone subsequence (Lemma 8.3.8, use your notes!!!), Bolzano-Weierstrass Theorem (Theorem 8.3.9), convergent sequences are Cauchy sequences (Theorem 8.3.12), Cauchy sequences are bounded (Lemma 8.3.14), Cauchy sequences are convergent (Theorem 8.3.15), Cauchy Completeness Theorem (Corollary 8.3.16, use Lemma 8.3.13 without proving it!), ratio of Fibonacci numbers converges to the golden ratio (Example 8.4.10), convergence and divergence of the geometric series (Example 9.2.4).
- Statements you should be able to state (without proof):
Archimedean property (Theorem 2.6.7), the real numbers are unique up to an order preserving isomorphism (Theorem 2.7.1), the limit does not change if we change finitely many entries in the sequence (notes), part (3) of Lemma 2.8.5, Squeeze Theorem (Theorem 8.2.12), a Cauchy sequence containing a convergent subsequence is convergent (Lemma 8.3.13), rephrase Theorem 8.2.9 as statements on series (Theorem 9.2.6, know which parts!)
- What to expect
The exam will be closed book. I will provide a list of logical implications and equivalences listed in Fact 1.3.1 and 1.3.2. You will have 80 minutes to answer about 10 questions. Some questions may ask you to state and prove a theorem from the list above, others may be like the exercises from your homework assignments. Even if a statement is listed "without proof" above, you must remember the proof of those parts of it that were on a homework assignment!