Some facts for Final Exam

Preface

Basic Knowledge Base for the Final

Here is a short version giving some things more emphasis.

• Technique of Mathematical induction
• The Completeness Axiom
• Definition of Inf S and Sup S. Be able to determine the inf and sup for concrete examples.
• Definition of convergence to a limit of a sequence (both as a finite number and as plus or minus infinity)
• Definition of a bounded (above, below or both) set and sequence
• The Archimedean principle
• Definition of the limsup(s_n) and liminf(s_n)
• Definition of a Cauchy Sequence
• Definition of convergence of an infinite series
• Cauchy Criterion for infinite series
• Definition of absolute convergence of an infinite series
• Convergence Theorem for Bounded Monotone Sequences
• Theorem about the equivalence of a "convergent sequence" and a "Cauchy sequence".
• The Bolzano-Weierstrass Theorem
• The Comparison, Ratio, Root, Integral and Alternating Series Tests
• Definition of Uniform Continuity. Be able to use the definitions of continuity, uniform continuity
• Theorem about the existence of a max and min for a real valued function which is continuous on a closed and bounded set (interval).
• Intermediate Value Theorem
• Theorem about Continuity on a closed and bounded interval (set) implying uniform continuity.
• Preservation of Cauchy sequences by uniformly continuous functions
• Be able to determine the continuity and uniform continuity for concrete functions
• Convergence of Power Series; radius of convergence, how to deal with the endpoints of the interval of convergence, the type of convergence (absolute!, uniform on closed intervals) inside the radius of convergence
• Convergence of a sequence of functions: definition of pointwise convergence, definition of uniform convergence.
• Theorem that the uniform limit of a sequence of continuous functions is continuous.
• Theorem that a limit can be taken inside the integral for uniform convergence.
• Definition of a uniformly Cauchy sequence of functions
• Theorem that a uniformly Cauchy sequence of functions is guaranteed to converge uniformly to a function.
• Definition of uniformly convergent series of functions.
• The Weierstrass M-test for the uniform convergence of series of functions.
• The Theorems that the term-by-term integration or the term-by-term differentiation of power series gives power series with the same radius of convergence and which are the integral or derivative (respectfully) of the function represented by the original series.
• Definition of the derivative. Differentiable implies continuity, Rolle's Theorem, Mean Value Theorem.