### 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 ${1}$.

However, there are ways of summing Grandi’s series. For most of these methods, the assigned value is ${G=\textstyle{\frac{1}{2}}}$. If we group the terms of Grandi’s series we can get two conflicting results. Grandi grouped the terms as follows:

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

However, he 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.

Summing Grandi’s Series

The Cesaro sum of a series is the limit, if it exists, as ${n\rightarrow\infty}$ of the arithmetic mean of the first ${n}$ partial sums. For Grandi’s series, this gives the value ${{\textstyle{\frac{1}{2}}}}$ (details in previous post).

But let us consider the series

$\displaystyle 1 + 0 - 1 + 1 + 0 - 1 + 1 + 0 - 1 + \dots$

where a ${0}$-term is inserted after each ${+1}$-term. The sequence of partial sums is

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

The sum of the first ${3n}$ terms is ${2n}$, so the mean is ${\frac{2}{3}}$. Thus the series, which is just Grandi’s series with some zeros added in, has Cesaro sum ${\frac{2}{3}}$. For the series

$\displaystyle 1 - 1 + 0 + 1 - 1 + 0 + 1 - 1 + 1 + \dots$

where a ${0}$-term is inserted after each ${-1}$-term, the sequence of partial sums is

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

The sum of the first ${3n}$ terms is ${n}$, so the Cesaro sum is ${\frac{1}{3}}$. These paradoxical results were discussed by Daniel Bernoulli.

Historical Interest

Grandi’s series was a subject of great interest for many famous mathematicians. Leibniz studied it, favouring the assignment of the value ${\textstyle{\frac{1}{2}}}$ to the sum. He believed that the argument from the binomial expansion of ${1/(1+x)}$ was valid. The limit of the expansion as ${x\nearrow 1}$ is Grandi’s series. while the limit of ${1/(1+x)}$ is ${\textstyle{\frac{1}{2}}}$.

Many other mathematicians discussed the series, including Jacob Bernoulli, Jean le Rond d’Alembert and, of course, Leonhard Euler. Euler argued that the series ${G}$ and the value ${\textstyle{\frac{1}{2}}}$ are `equivalent quantities’, and that it is always permitted to substitute one for the other without error. Euler manipulated divergent series with great effect, although sometimes with confusing and contradictory results.

Augustin-Louis Cauchy gave a rigorous definition of the convergence of series and this led, for some time, to the elimination of divergent series from most mathematical deliberations. They reappeared around 1886 as a result of Henri Poincaré’s work on asymptotic series. Divergent series now play a central role in many areas of both pure and applied mathematics.

Sources

${\bullet}$ Wikipedia article History of Grandi’s Series:

${\bullet}$ Wikipedia article Divergent Series: