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’Hôpital’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’Hôpital’s rule.
• Applications: Definition of exp, log, sin, cos, sinh, cosh.
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