Review Topics for Final Exams
Review Topics for Exam 2
Sequence and Series Basics: Definition of convergence and divergence; knowing the difference between sequences and series regarding convergence and divergence; nth-partial sums, absolute convergence, conditional convergence, interval and radius of convergence for power series, Taylor’s polynomials and Taylor’s series.
Sequence: techniques for finding limits for rational sequences; leading order estimation technique; how to determine a sequence is monotone, bounded; limits of monotone and bounded sequences; squeeze theorem;
Convergence/Divergence Tests for Series: nth-term test for divergence; integral test, basic comparison test, limit comparison test, p-series, ratio test, root test, alternating series test; absolute convergence implies convergence; Taylor’s theorem.
Remainder/Error Estimations: Error estimate by integral test; by alternating series test; by Taylor’s remainder formula.
Elementary Series and Taylor Series: Geometric series, telescoping series, p-series, Taylor’s series for exponential function, sine and cosine function, tangent inverse, logarithmic function.
Techniques: Deriving new series from known ones: add/subtract; multiply/divide; differentiate/integrate.
Application of Taylor Series: Finding limits; approximating integrals.
Miscellaneous: elementary sequences of converging and diverging types from page 509, Taylor series from page 571, convergence of telescoping series, formulas for both finite and infinite sums of a geometric series, integration by parts, by substitution, convergence and divergence of improper integrals
Partial Review Problems: All homework, quiz, lecture examples, and sample exams.
Review Tips: If you usually find yourself short on time when taking exams redo some problems may help. These problems may include those from your textbook, my lecture, and your homework and quizzes. Redo them neatly, completely, and do so until you can get them right without referring to their solutions. Only after you have done a thorough review of the subject and redo of these problems that you want to work out some sample tests and new problems.
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