mathematical dialogues

5 January 2022
A bumpy ride
We take again inspiration from the Putnam competition, for an exercise on continuity and differentiability of real fuctions obtained by "stretching" a basic replica. Let...
Leggi di più - Read more
5 December 2021
Problem unraveling
Let us go back to Terence Tao's Solving Mathematical Problems, for another exercise. As done in a previous post, I would like to show an alternative solution to a problem...
Leggi di più - Read more
2 December 2021
Extreme derivatives
Theorem. Let \(f\) be a real function, which is differentiable up to order \(n\) in  \(c \in \Bbb R\), for some positive integer \(n\). Suppose that \(f^{(n)}(c) \neq 0\), and...
Leggi di più - Read more
30 October 2021
Euclid strikes back
I found the following problem on Terenece Tao's "Solving Mathematical Problems". I propose, here, a synthetic approach based on simple Euclidean geometry concepts, instead of...
Leggi di più - Read more
17 July 2021
Close enough?
While solving some exercises for my students, I bumped into the following sub-problem: given a divergent sequence \((a_n)\), such that \((a_{n+1} - a_n)\) converges to \(1\), can...
Leggi di più - Read more
25 September 2020
The power of powers
In this post we make use of Taylor polynomial expansions to efficiently solve one of 2018 Putnam competition problems. Here is the problem statement. Let \(f:\Bbb R \to...
Leggi di più - Read more
9 July 2020
Uneven integration
Very seldom does one need to make use of the actual definition of Riemann integral. Most of the times, in fact, we are dealing with very "regular" functions for which a whole...
Leggi di più - Read more
20 January 2020
Euler's number \(e\), the base of the natural logarithm, manifests itself in various mathematical contexts, sometimes unexpectedly. In the following you will read an example of...
Leggi di più - Read more
23 December 2019
Analysis and synthesis
While solving some geometrical problems, I have been investigating on the possible alternatives between a trigonometric approach versus an approach based on congruences and...
Leggi di più - Read more
25 November 2019
Breaking the harmony
I found this curious exercise on the book Problems and Theorems in Analysis I, by Pólya and Szegö, and I thought it was a nice problem to propose here, since its solution...
Leggi di più - Read more
14 November 2019
Non sequitur
In this post the behavior of sequences obtained by convolving two sequences is analyzed. In particular, we will give sufficient conditions for the convolution to be a null...
Leggi di più - Read more
22 October 2019
The revenge of Menelaus
This post is devoted to Menelaus's Theorem, named for the astronomer Menelaus of Alexandria. It is a very imporant theorem in geometry, relating the ratios of the segments...
Leggi di più - Read more
10 October 2019
My slant on asymptotes
In this post we will investigate the connection between slant asymptotes and the behavior of the derivative at \(\infty\) for differentiable real functions. We say that a...
Leggi di più - Read more