In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series
This is a divergent series: the sequence of partial sums is 
Posts Tagged 'Analysis'
Grandi’s Series: A Second Look
Published July 26, 2018 Occasional Leave a CommentTags: Analysis, History
Grandi’s Series: Divergent but Summable
Published July 12, 2018 Occasional Leave a CommentTags: Analysis, History, Puzzles
Is the Light On or Off?
Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by
Fourier’s Wonderful Idea – II
Published April 5, 2018 Irish Times Leave a CommentTags: Analysis, History
Solving PDEs by a Roundabout Route
Joseph Fourier (1768-1830)
Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].
Fourier’s Wonderful Idea – I
Published March 29, 2018 Occasional Leave a CommentTags: Analysis, History
Breaking Complex Objects into Simple Pieces
“In a memorable session of the French Academy on the
21st of December 1807, the mathematician and engineer
Joseph Fourier announced a thesis which inaugurated a
new chapter in the history of mathematics. The claim of
Fourier appeared to the older members of the Academy,
including the great analyst Lagrange, entirely incredible.”
Introduction
Joseph Fourier (1768-1830)
The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?
Subtract 0 and divide by 1
Published March 8, 2018 Occasional Leave a CommentTags: Analysis, History
We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that
This is an essential step in deriving an expression for the derivative of 
The Evolute: Envelope of Normals
Published February 22, 2018 Occasional Leave a CommentTags: Analysis, Geometry
Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.
Sin t (blue) and its evolute (red).
Torricelli’s Trumpet & the Painter’s Paradox
Published April 13, 2017 Occasional Leave a CommentTags: Analysis, Geometry, Recreational Maths
Torricelli’s Trumpet
Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve 
Continue reading ‘Torricelli’s Trumpet & the Painter’s Paradox’
