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Program

Updated A.Y. 2019-2020

 

Set theory. Set Operations: union, intersection, difference, complement, Cartesian product. Number sets. Intervals. Interior, exterior and accumulation points.  Maximum, minimum, inferior and supremum. Powers with rational and real exponents. Funcions: domain, range, image. Injective and surjective functions. Inverse function.

Real valued functions: linear, quadratic and polynomial functions. Exponential, Logarithmic and  Trigonometric functions. Monotonic functions.

 

Sequences: definition of limit of a sequence, uniquness of the limit of a sequence, monotonic sequences, bounded sequences. Existence of the limit of a monotonic and bounded sequence.
Recursively defined sequences: definition and computation of the limit. Economic application: a dynamical model of demand and supply in discrete-time. Derivation of the notable limits n*sin(1/n) and (1+1/n)^n. The Euler Number. Irrationality of the Neper Number (just stated but not formally proved).

Series: formal definition. Derivation of the necessary condition for convergence. The root test. The ratio test. The condensation criterion. Convergence of the armonic series. Convergence of the geometric series. Summability of telescopic series. The limit comparison test for Series. The convergence of the series of the reciprocal of factorials to the Neper number.