In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. Mathematical analysis is largely taken up with studying the conditions under which a given function will result in a convergent infinite series. Such series (e.g., the Fourier series) are particularly useful in solving differential equations.
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Let us say, therefore, that the sum of any infinite series is the finite expression, by the expansion of which the series is generated. In this sense the sum of the infinite series 1 − x + x2 − x3 + · · · will be 1⁄1+x, because the series arises from the expansion of the fraction, whatever number is put in place of x. If this is agreed, the new definition of the word sum coincides with the ordinary meaning when a series converges; and since divergent series have no sum, in the proper sense of the word, no inconvenience can arise from this new terminology. Finally, by means of this definition, we can preserve the utility of divergent series and defend their use from all objections.|30px|30px|Euler|1755