Partial sum

In mathematics, a series is, roughly speaking, a description of the operation consisting of adding, one after the other, infinitely many quantities to a given starting quantity. For a long time, the idea that such a potentially infinite summation may have sense and may produce a finite result was considered paradoxical by mathematicians and philosophers, and only by the end of the 19th century was this paradox fully resolved using the concept of a limit.

Zeno's paradox of Achilles and the tortoise illustrates the couterintuitive property of infinite sums: Achilles runs afters a tortoise; 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. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. In fact, Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time and could not accept that the total time of all these sub-races can be finite.

In modern terminology, any (ordered) infinite sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ of terms (that is numbers, functions, or anything that can be added) defines a series which is the operation of adding the ${\displaystyle a_{i},}$ one after the other. To emphasize that there are an infinite number of terms, a series is often called an infinite series. Such a series is represented (or denoted) by an expression like

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