MPU1054 – Analysis in One Real Variable II

Fall 2026 · Université du Québec à Trois-Rivières · Department of Mathematics and Computer Science · taught in French

This course develops the theory of the Riemann integral and deepens the understanding of integral calculus and of series of functions: uniform continuity, the Riemann integral (fundamental theorem of calculus, improper integrals), sequences of functions, numerical series, series of functions and power series (Taylor series). It follows MPU1045 – Analysis in One Real Variable I, which is its prerequisite.

The course has four chapters:

1. the Riemann integral;
2. sequences of functions;
3. numerical series;
4. series of functions.

Applications

Interactive applications to explore the notions of the course. They run directly in the browser, on a phone as well as on a computer, and can be projected in class.

• Riemann and Darboux sums (Chapter 1) – partitions, tags, Riemann sums, lower and upper Darboux sums, refinement and the limiting process.
• Pointwise and uniform convergence (Section 2.1) – the pointwise limit, a few members of the sequence and the band of width \( 2\varepsilon \) around the limit, for the examples from the notes, the exercises of Section 2.1 or a sequence of your own: the convergence is uniform when, from some rank on, the graph of \( f_n \) lies entirely inside the band.

Activities

Proofs and examples from the course to complete by choosing, for each blank, the right option from a menu, with immediate feedback and a printable version for the course notes. The activities are in French.

• Fill-in-the-blanks proof – Corollary 1.34 (Section 1.6) – the proof that \( |f| \), \( f^2 \) and \( fg \) are integrable.
• Fill-in-the-blanks example – the sequence \( x^n \) on \( [0, 1] \) (Section 2.1) – the pointwise limit case by case, the continuity of the limit and the rank \( N(\varepsilon, x) \), which depends on \( x \): pointwise convergence does not preserve continuity.

“Voir la solution” (show the solution) only appears after a few attempts: try by yourself first.