Sequences math definition1/19/2024 ![]() In modern terminology, any (ordered) infinite sequence ( a 1, a 2, a 3, … ) at the origin and converges to it for every x. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. ![]() Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position when he reaches this second position, the tortoise is at a third position, and so on. This paradox was resolved using the concept of a limit during the 17th century. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.įor a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. ![]() The study of series is a major part of calculus and its generalization, mathematical analysis. In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
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