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Unformatted text preview: Mat 315 Review for Exam 2 February 25, 2008 The first exam is Monday, March 3. The exam will cover sections 6.2-6.6. You should study your homework assignments, the notes for the class and the examples and proofs in the book. You definitely should know the definitions of the following terms: fn f pointwise fn f uniformly on A fn converges pointwise to f fn converges uniformly to f power series radius of convergence Also know the following theorems: Criterion for Uniform Convergence ( )Theorem 6.2.6: The uniform limit of continuous functions is continuous Theorem 6.3.1: Hypotheses that guarantee that (lim fn (x)) = lim fn (x) ( ) Theorem 6.4.2: The uniform sum of continuous functions is continuous. Theorem 6.4.3: Hypotheses that guarantee that ( fn (x)) = fn (x) Theorem 6.4.4: Cauchy Criterion for Uniform Convergence of Series ( ) Corollary 6.4.5: Weierstrass M -Test ( )Theorem 6.5.1: A power series which converges at one point is absolutely convergent on some interval. Ratio Test Theorem 6.5.5: A power series which converges on an interval must converge uniformly on any compact subset of that interval. Theorem 6.5.6: If a power series converges on some open interval, then d dx for all x in that open interval. Theorem 6.5.7: Summary of the wonderfulness of power series LaGrange's remainder theorem an xn = d (an xn ) = dx nan xn-1 The exam will have: One question that asks for statements of theorems or definitions One question that is very similar to homework problems One question that asks you to prove one of the starred theorems from this review sheet One question that asks you for examples. Some more questions ...
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- Winter '08