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ANALYSIS I and II 1. Sequences. • Definition of convergence. Finite and infinite limits. Allowed operations. • Test for convergences: Comparison (Sandwich), Cauchy, Ratio. Useful examples: √ √ P (n) an , n a, n n, nn!n and Q(n) where P, Q are polynomials. 2. Real numbers. • Basic properties. Rational and irrationals are dense (intersect every open interval). • More important to you: Axiom of completeness in all its forms. “Bounded below ⇒ infimum”, “bounded above ⇒ supremum” and “Every Cauchy sequences converges”. 3. Series. • Translate the results from sequences. Tests for convergence: Comparison (Boundedness), Ratio and Integral. P n P n! P1 P 1 • Useful series. , with p > 1, a , . n np nn • Alternating series test. Absolute and conditional convergence. 4. Continuous Functions. • Definition. Equivalent formulation in terms of sequences lim f (·) = f (lim ·). Allowed operations. Finite and infinite limits. • Two important consequences of continuity: Intermediate value and Min-Max theorems. 5. Differentiability. • Definition. Diff ⇒ Continuity, but not the other way around. Allowed operations, Leibniz and Chain Rules. • Important consequences of differentiability: Finding Min-Max of a function, Rolle’s and Mean value theorems, f 0 > 0 ⇒ f increasing. L’Hò‚pital’s rule, first version. 6. Higher order derivatives. • Definition. • Power series. Ratio, root, limsup and liminf tests. Ratio of convergence. • Taylor’s theorem. Estimating errors. More l’Hò‚pital’s rule. • Applications: Definition of exp, log, sin, cos, sinh, cosh. 1