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.


Grandi’s Series

An infinite series, the sum of an infinite number of terms, may or may not converge to a finite limit. The geometric series

\displaystyle S = {\textstyle{\frac{1}{2}}} + ({\textstyle{\frac{1}{2}}})^2 + ({\textstyle{\frac{1}{2}}})^3 + \dots

converges to the limit 1. To show this, we calculate the sum of the first {n} terms, called the partial sum, and consider the limit of the sequence of partial sums. The {n}-th partial sum is

\displaystyle S_n = {\textstyle{\frac{1}{2}}} + ({\textstyle{\frac{1}{2}}})^2 + ({\textstyle{\frac{1}{2}}})^3 + \dots + ({\textstyle{\frac{1}{2}}})^n = {\textstyle{\frac{1}{2}}} \displaystyle{\left(\frac{1-(1/2)^{n}}{1-(1/2)}\right)} \longrightarrow 1


Dom Guido Grandi (1671 – 1742)

A notorious series that does not converge was studied by the Italian monk Dom Guido Grandi around 1703. In 1707 Grandi was appointed court mathematician to the Grand Duke of Tuscany, Cosimo III de Medici. Grandi studied both Newton’s fluxions and Leibniz’s differentials and sent copies of his work to both these mathematicians. He received thanks from Leibniz and Newton sent him copies of his Opticks and Principia. In 1709 Grandi was elected a Fellow of the Royal Society in London, having been proposed by Newton. Grandi studied geometric curves including one now known as the Witch of Agnesi. The University of Pisa appointed Grandi as Professor of Mathematics in 1714.

Grandi is probably best-remembered today for the divergent series

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

It is clear that the sequence of partial sums of this series is

\displaystyle \{ 1, 0, 1, 0, 1, 0, \dots \}

which obviously does not converge, but alternates between {0} and {1}.

This is the sequence of states in the light puzzle posed above. The question of whether the lamp is on or off is equivalent to the question “What is the sum of Grandi’s series?”

What is the Sum?

Can we assign any meaning to the infinite sum? Grandi used the binomial expansion

\displaystyle \frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots

(which is valid for {|x|<1}) and substituted the value {x=1} to get

\displaystyle \frac{1}{2} = 1 - 1 + 1 - 1 + \dots

implying that the sum of his series is {{\textstyle{\frac{1}{2}}}}. But of course there are difficulties with this approach.

If we group the terms of Grandi’s series we can get two conflicting results. Grandi grouped the terms as follows:

\displaystyle G = ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + \dots = 0 + 0 + 0 + \dots

This led him to argue that the sum of an infinite number of zeros equals {{\textstyle{\frac{1}{2}}}}. However, he might just as well have concluded that the sum of his series is {0}.

Grandi might also have regrouped as follows

\displaystyle G = 1 - ( 1 - 1 ) - ( 1 - 1 ) - ( 1 - 1 ) - \dots = 1

giving a sum of {1} for the series. Of course, we now understand that re-grouping of the terms of a divergent series is not justifiable, and may yield nonsensical results.

Somehow, Grandi’s initial argument that the sum is {{\textstyle{\frac{1}{2}}}} seems intuitively appealing. There is another way to “demonsrate” this:

\displaystyle 1 - G = 1 - ( 1 - 1 + 1 - 1 + \dots) = 1 - 1 + 1 - 1 + 1 + \dots = G

which gives {1-G = G} or {G = {\textstyle{\frac{1}{2}}}}.

Cesaro Sum of the Series

Cesaro summation, named after the Italian mathematician Ernesto Cesaro (1859–1906), may sometimes be used to assign a value to a divergent infinite series. The Cesaro sum of a series {a_1+a_2+a_3+\dots} is the limit as {n\rightarrow\infty}, if it exists, of the arithmetic mean of the first {n} partial sums {s_n=a_1+ \dots +a_n}, that is,

\displaystyle \lim_{n\rightarrow\infty} \frac{s_1+s_2+\dots +s_n}{n}

For Grandi’s series, the partial sums are {\{1,0,1,0,1,0,\dots\}}. The average of the first {2n-1} terms is {n/(2n-1)} and of the first {2n} terms is {1/2}. That is, the sequence of averages is

\displaystyle \left\{ \frac{1}{1}, \frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6}, \dots \right\}

which has the limit {{\textstyle{\frac{1}{2}}}}. Thus, the Cesaro sum of Grandi’s series is {{\textstyle{\frac{1}{2}}}}.

Does this shed any light on the lamp in the puzzle above? Not really; unless we conclude that, like Schrödinger’s Cat, the lamp is both on and off, with equal probabilities.


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