Convergence and Limit

Abstract

Central to the development of the calculus were the concepts of convergence and limit, and with these concepts at hand it became at last possible to resolve the ancient paradoxes of infinity which had so much intrigued Zeno. For example, the runner’s paradox is explained by the following observation: By first covering one-half the distance between the runner’s starting and end points, then half the remaining distance, and so on, he will cover a total distance equal to the sum:

$$ 1/2 + 1/4 + 1/8 + 1/16 + \cdots $$

A quantity is the limit of another quantity, when the second can approach the first more closely than any given quantity as small as one can suppose.

— Jean Le Rond D’Alembert (1717–1783)