Real Analysis II

Syllabus

• Derivative and differentiability; differentiation rules and derivatives of elementary functions; higher-order derivatives.
• Local behavior vs derivative; mean value theorems; qualitative analysis of differentiable functions.
• Taylor formula; l’Hôpital’s rule.
• Antiderivatives and integration techniques: substitution, integration by parts; integrals of rational functions.
• Riemann integral: definition, integrability conditions, basic properties, and estimation of integrals; Newton–Leibniz theorem.
• Applications of integration: Wallis formula, Stirling formula; Taylor formula with integral remainder.
• Improper integrals: definition, convergence conditions; examples of integrals without elementary antiderivatives.
• Functions of bounded variation; Riemann–Stieltjes integral; second mean value theorem for integrals.
• Infinite series: absolute convergence; major convergence tests (comparison, root, ratio, integral, alternating/Leibniz, Abel–Dirichlet).
• Products and rearrangements of series: Cauchy product, Mertens theorem (overview), rearrangements and Riemann’s theorem (statement).