Category: Exercises and dialogues

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...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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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...
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23 September 2019
Yet another triangle
As we have seen in another post, identities involving inverse tangent look much less mysterious if analyzed from a purely geometrical perspective. Here you have the chance to...
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26 July 2019
Feeling grilled?
Get pen and paper and draw a cartesian plane. Then draw, on it, a set of points with integer coordinates, such as \(A(1,1)\), \(B(1,0)\), \(C(2,0)\), \(D(3,2)\), and so forth......
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10 July 2019
The bare necessities
This is the first of a series of posts in which a problem is presented to the reader, who in invited to solve it by following a proposed path. The background knowledge required...
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