1826.] tJie Use of continued Fractions, S)C, 419 



which will, therefore, consist of a continual alternation of terms 

 similar to the last two. The collected result is, 



a 



ffl = — a, 



U + &c (3) : 



The continued fraction is exactly the same as would have 

 resulted from performing the like operation on the series itself; 

 but there is a manifest advantage in having all the elements of 

 convergency in one view. 



3. When any of the quantities ip becomes = 0, the fraction is 

 terminated, and the series is truly summed. E. g. l{<p. = .-. 

 g, = 0, and not s, only, but e^, £3, 8cc. since p, a^, a., &c. are to 



be left perfectly arbitrary. Whence ip^ = -^, the other terms all 



vanishing, and that individually, viz. yi p—y.^ o^ = 0, 5", p — 3|j a^ 



= 0, &c. on the same principle. Therefore f, = ^ is to be 



^, + — 



V. 



identical with —^^, — r, the remaining terms vanishing, 



and separately as before. In the same manner % = \ +'f _2p 



1 - (p - 02) 1 + p + Q m 



0,= —^ ^, <P = -i • Ihe remaiumg 



T^' 1 + (p - a,) + (q — OtP + a, a J ^ i + p + q = 



quantities R, S, &c. r, s, See. are therefore each = 0, and their 

 equations reduce to 



q — a,p + a, a^ - 



9 — 03 p + «3 a^ = 



q — a.p + ff, o, = &c (4) 



Conversely, if the series be recurring, the fraction will termi- 

 nate, as will appear by reversing this reasoning, since the sum 

 of every such series is expressible by a finite portion of 



1 + P + Q + .... 



a. 



1 + p + q + 



If the series do not terminate, the continued fraction never- 

 theless affords an approximation towards the value of the series, 

 and means of estimating the degree of approximation, viz. either 

 from the law of continuation of the fraction itself, or by the 

 equations for P, Q, R, &c. 



While the quantities <p are all affirmative, the fraction is cer- 

 tainly convergent toward the correct value of the series. 



By the aid then of the successive formulaj in Art. 2, any series, 

 but particularly such us are distinguished by alternating signs, 

 may be summed or algebraically approximated by continued 

 fractions, and in general arithmetically too, without the tedious 

 process of direct division. 



4. In illustration of these general remarks, let it be proposed 



2 e2 



