SERGEANT

Sec Serjeant. SERIES (Lat. order), in Arithmetic and Algebra, is a progre ion of numbers or quantities which succeed each other according to some determinate law. Thus, the progre ion of the natural numbers, 1, 2, 3,4, 5, , constitutes a series; the law of which is, that any term is greater by unity.than the preceding. As the law which connects the terms may be varied in an infinite number of ways, it follows that series may have an infinite number of different forms; thus, an arithmetical series is one in which each term differs from the preceding by a constant number or quantity; a geometrical series is one to which each term is a multiple of the preceding by a constant factor. The usual form of a series is a set of terms connected by the signs -4- or —. When the number of terms is greater than any a ignable number, the series is said to be infauu. A converging scries is one in which the succe ive terms become le and le , each being smaller than that which immediately precedes it. A diverging scries is one in which any term is greater than the preceding. A recurring scries is one in which each term is a certain constant function of two or more of the preceding terms Thus, 1 + 3 x -f- 4 x2 + 7 x3 + 11 %\ + ., is a recurring series, every term being formed in the same manner from the two immediately preceding. This series is formed by the expansion by division of the algebraic fraction l + 2z 1 — x — xi An exponential scries is one whose terms depend on exponential quantities; a logarithmic series is one whose terms depend on circular functions, as sines, cosines, . The general term of a series is a function of some indeterminate quantity x, which, on substituting succe ively series.the numbers 1, 2, 3, , for x, produces the terms of the In general, the principal questions which arise respecting series are to find the sum of all the terms, or of a given number of them; and the exposition of the methods which have been invented for this purpose forms an extensive and important part of every treatise on the calculus. The student may be recommended in particular to Filler's Introductio in Jinalysin In jin i to rum, and Calculus Differentials; and the third volume of Lacroix's Calcul Differential ct Integral. [s. 1126]