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Lecture 2: Exercises

• Convergence tests: Arfken (chaps. 5.2, 5.3)

Questions for Review

• Which are the most important convergence tests for infinite series?
• What is the comparison test?
• What is the Cauchy root test?
• How is the Cauchy root test related to the comparison test?
• What is the D'Alambert ratio test? How is it related to the comparison test?
• What is the Leibnitz criterion? To which kind of infinite series does it apply? Give an example!
• How is Euler's number e defined? What is (approximately) its numerical value?
• How can functions be represented?
• What does it mean to say that a function is
• bounded?
• positive or negative definite?
• even or odd?
• periodic?
• (strictly) monotonic?
• How is the inverse of a function defined?
• How can the inverse be constructed graphically?
• Do all functions have an inverse? Give examples!
• Do all strictly monotonic functions have an inverse?
• Are all invertible functions strictly monotonic?
• Are the functions x^2 and cos(x) invertible? If so, why? If not, what can one do in order to make them invertible?
• When is a function continuous?
• Which are the most common types of discontinuities?
• What is the Heaviside step function? Is it continuous?
• Can functions have a well-defined limit at a gap? If so, how is this formulated mathematically?
• What is the limit of sin(x)/x as x goes to zero?
• Give examples for (nontrivial) functions that approach an asymptotic value as the argument goes to plus or minus infinity. How is this formulated mathematically?