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7 and 118 sample problems 3 series definitions

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of subsequential limits (11.7 and 11.8). Sample problems: 3. Series. Definitions: convergent series, absolute convergence, Cauchy property for series. Theorems: sum of a geometric series, absolute convergence implies convergence, comparison test, ratio test, root test, integral test, alternating series test. Sample problems: basically you need to be able to use all of these tests and the two standard series ( 1 /n s and 1 /a n ) to prove that a given series converges or diverges. There’s a large number of problems in the book. 4. Continuous functions and their properties. Definitions: continuous function. Theorems: ε - δ definition of continuity (17.2), sums, prod- ucts, ratios and compositions of continuous functions are continuous (17.3-17.5), a continuous function on a closed interval is bounded and has extrema (18.1), intermediate value theorem (18.2). 5. Uniform continuity. Definitions: uniformly continuous function. Theo- rems: a continuous function on a closed interval is uniformly continuous (19.2), Cauchy sequences (19.3), uniformly continuous functions extend 1
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(19.5). Problems: prove that a given function is or is not uniformly continuous (19.1-19.3 and 19.5-19.6). 6. Limits. Definitions: limit of a function along an arbitrary approach set S (20.1) and its various specializations (20.2, 20.3). Theorems: limit theorems (20.4 and 20.5) ε - δ definition of a limit (20.6), and the various specializations (20.7-20.9), left and right limits versus limit (20.10) Problems: you need to be able to write down the two different definitions of a limit (using subsequences and ε - δ ) for any of the 15 different possibilities (limit at a finite point, left and right limits, limits at infinity, and infinite limits). Discussion 20.9 is very useful here. Also, you need to be able to prove that a given function has a given limit at a point (from the definitions directly or using any of the limit theorems). 2
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