Università Roma Tre

REAL NUMBERS AND OTHER INTRODUCTORY SUBJECTS
The rationals; $\sqrt 2$ is irrational. The real numbers contain the rationals and have the sup property. A distance on the reals; functions and sequences; limits of sequences. Monotone sequences always admit a limit; series with positive terms; the comparison test, the condensation test, the root and ratio test. Cauchy sequences; series with non-positive terms; absolute convergence and convergence. The Leibnitz convergence theorem. power series and convergence radius. Definition of sine, cosine, exponential and logarithm through their series; important limits.

REAL FUNCTIONS
Definition of the limit of a function; continuous functions; power series are continuous; uniform continuity; a continuous function in a closed and bounded interval is uniformly continuous. The intermediate value theorem; the theorem of Weierstrass. The derivative; differentiability implies continuity; the derivative in the point of maximum is zero. The chain rule and the derivative of the inverse. The theorems of Rolle and Lagrange. Convex functions; the slope of their chords is increasing; if they are differentiable, the derivative is increasing. Inequalities among the means and Young's inequality. Two versions of L'Hopital's theorem. Taylor's formula with the remainder in the forms of Lagrange and of Peano. Convergence of Taylor series: counterexamples and examples. The relationship between convergence of sequences and of functions.

INTEGRATION
Riemann's integral; integration by parts and by substitution; fundamental theory of calculus. The complex numbers; the complex plane and Euler's representation. Integration of rational functions. Taylor's formula with integral remainder. Improper integrals; absolute integrability. Uniform convergence for sequences and series of functions. Limits of Riemann integral under uniform convergence. Power series can be integrated and differentiated term by term.

NERDY THINGS
Various approximations of $\sqr 2$; machin's formula for $\pi$; the infinite product of Wallis; the formula of De Moivre Stirling; the area under the Gaussian with the formula of Wallis; Euler's summation formula; a model of the distribution of wealth; a continuous but nowhere differentiable function.

B. Palumbo - M.C. Signorino, Funzioni algebriche e trascendenti, ed. Accademica, 2015.

Marcellini-Sbordone, Analisi Matematica 1.

De Marco-Mariconda, Analisi Matematica 1.