## Archive for July, 2018

### Grandi’s Series: A Second Look

In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series

$\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots$

This is a divergent series: the sequence of partial sums is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$, which obviously does not converge, but alternates between ${0}$ and ${1}$.

### The Empty Set is Nothing to Worry About

Today’s article is about nothing: nothing at all, as encapsulated in the number zero and the empty set. It took humanity millennia to move beyond the counting numbers. Zero emerged in several civilizations, first as a place-holder to denote a space or gap between digits, and later as a true number, which could be manipulated like any other. [see TM143, or search for “thatsmaths” at irishtimes.com].

A selection of images of zero (google images).

### Grandi’s Series: Divergent but Summable

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 ${1}$ and ${0}$, the sequence of states over the first minute is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### Trigonometric Comfort Blankets on Hilltops

On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country  [see TM142, or search for “thatsmaths” at irishtimes.com].

Trig pillar on summit of Croaghan Moira, Wicklow [Image from https://mountainviews.ie/%5D.