SERIES. The mathematical of the word series is, a set of terms, finite or infinite in number, connected by addition or subtraction, and formed upon some distinct law. if it had been the plan of this work to write treatises on the various branches of pure mathematics, the present article would have been brief, and abounding in references to the articles on algebra and the differential calculus, the most important results of which arc expressed in but in a work which, without into such full details, professes to furnish references to the most important detached doctrines of the exact sciences, the present article must extend to some Series may be either finite or infinite in the number of their terms. As to finite series, such for instance as' x terms of 1+2 +3+ . • • • ; the only of importance which arises with respect to them is, how to express the sum as a function of the indefinite number of terms, s. On this point wo refer to the articles INTE GRATION, FINITE, and SUMMATION : it is with the doctrine of infinite series that the mathematician is more particularly concerned in the present article. as to the manner in which the differential calculus in applied to the development of functions in series, we refer to TAYLOR'S THEOREM.
A series of an infinite number of terms may be either purely nume rical, as 1+2+3+4+ in which the symbol + ..., or +,&c., means that the series is to be carried on for over, the law of formation of the written terms continued all the unwritten ones ; or it may contain literal expressions with an obvious law of formation, as in 1 +2.e + 3.r4+ .... A series of the latter clam is reduced to one of the former so soon as any definite value is given to the letters it contains.
An infinite series may be either or as explained in the article CONVEROE:4T. The various tests' there ex plained will perhaps serve to settle this point as to the number of series actually employed ; but the (`Diff. Cale," Lib. ol ecfnl Knowl./ pp. 236, 326; we shall refer to thin work in the ander the letters D. C.) will leave no doubtful case, its appli eation may eometimes be troublesome.
Let +x be the Ali term of a series *1 +* 2 ++3 + . ... (thus iS the xth term of I +a +0+ of I rind let —x4/x : sV x being the differential coefficient of ty.c. Let a, be the limit of ts, when x increases without limit ; then if a„ be greater than I, the series is convergent ; if a, be less than I included), divergent; if =I, doubtful.
In the doubtful case of the preceding, let r, =log x
and let a, be its limit when x increases without limit. Then if a, be >1, the series is convergent ; if
By the symbol Rex is here meant the series 40X + 4)(X + I) + 4)(X + 2) + ..; but when a number is written beneath s, as in sp it indicates the value of x in the first term of the series. Thus s,x stands for 4 + 5 +6 + , s, x stands for a + log (a + 1) + . Sonic, such abbreviation is most wanted in an article of reference, in which compression is but the student should write his series at more until he is well accustomed to them.
A series is, arithmetically that is, the by its terms may be made than any agreed on at the of the process. Such is evidently the case with 1 + 2 + 4 + ...., or s,,2 s. Nevertheless, as every knows, such series have been used as the representatives of finite quantities. It was usual to admit such series without hesitation ; but of late years many of the continental mathe maticians have declared series and have asserted instances in which the use of them leads to false results. Those of a contrary opinion have replied to the instances, and have from principles in favour of series. Our own opinion is, that the instances have arisen from a misunder or misuse of the series employed, sufficient to show that series should be very carefully but that, on the other hand, no perfectly and indisputable to the use of these series has been established d priori. They appear always to lead to true results when properly used, but no demonstration has been that they must always do so.