From the paradoxes to the rigorous limit

The mathematical tool that resolves Zeno’s paradoxes is called the limit. When the Greeks wrote that a geometric series with ratio $|r|<1$ has sum

$S = \frac{a}{1-r},$

and when Archimedes used the method of exhaustion to compute areas, they already knew the substance of the problem. What was missing was the formal language to say what it means that “infinitely many terms sum to a finite number”.

This language — the $\varepsilon$-$\delta$ definition of the limit — arrives with Cauchy and Weierstrass in the 19th century, and it is the starting point of the whole chapter (Boyer). For the context of Eudoxus’s and Archimedes’s method of exhaustion see also Netz.

Observation — A connection already seen

The particular case of geometric series of areas has already been met in Year Two: it is the concrete connection that we now generalise with the concept of the limit.

Links

Topics: Limits
Concepts: Limit · Geometric series
People: Archimedes · Augustin-Louis Cauchy · Karl Weierstrass