imprecise notion of taking the sum of an endless sequence of numbers. The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers. The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms. Instead, the finite sum of the first terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as grows without bound. The series is assigned the value of this limit, if it exists. Given an (infinite) sequence , we can define an associated series as the formal mathematical object sometimes simply written as . The partial sums of a series are the numbers . A series is said to be convergent if the sequence consisting of its partial sums, , is convergent; otherwise it is divergent. The sum of a convergent series is defined as the number The word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms. For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the Riemann rearrangement theorem for further discussion). An example of a convergent series is a geometric series which forms the basis of one of Zeno's famous paradoxes: In contrast, the harmonic series has been known since the Middle Ages to be a divergent series: (Here, "" is merely a notational convention to indicate that the partial sums of the series grow without bound.) A series is said to converge absolutely if is convergent. A convergent series for which diverges is said to converge non-absolutely. It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a series that converges non-absolutely is Taylor series The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power series : which can be written in the more compact sigma notation as