1826.] the Use of continued Fractions, S^c. 417 



]fx=A-B + C-D +E-F+&C. 



A 



Then is a; = y- ^ ^ a c 



A-B + 



BD 



B-C + TT-FT CE 



C-D + 



D-E + &c. 



By means of this theorem, a multitude of series may be turned 

 into forms, distinguished for elegance, and convenience to the 

 memory, as any one may convince himself by trial with the 

 common circular, logarithmic and other series. Ihe great 

 drawback on the score of utiUty is this, that the fraction and the 

 series it represents, proceed pari passu; the aggregate ot any 

 number of terms of the one, being the same as that ot an equal 

 number of terms of the other. So that the fraction is merely a 

 transformation of the series, and not an expression for its sum. 



To ffive a single instance : let this be the hypergeometnc 

 series 1-2 +6—24 + 120 - &c. Turned into a continued 

 fraction by the above formula, it becomes 



- 2 



—4 + ■ 



-18 H- &c. 



Or, every where dividing ttoo successive numerators and the 

 denominator belonging to the first of them, hy a common measure 

 of the three,* this formula becomes 



-3 + &e. 



the law of continuation being manifest. 



Here 1 is become 5-, 1 -2 is become ^ ^ 2 and so forward. 



ZTi' 

 But as regards the value of the series, we only discover that the 

 successive links of the continued fraction converge very evidently 

 toward the limit -i^, which, abstracting the sign, is always too 

 small ; that consequently the aggregate of m successive links is 

 still more nearly = -^j, which abstracting the sign is too small 

 or too great, as m is odd or even ; and that consequently the 

 whole series has for its limits J and =y- ; a conclusion to which 



the series itself would have conducted us equally well. 



2. In Prob. 51 of the Med. Alg. Waring has remarked that 



• If this simple mode of reduction had occurred to Euler, he would not, I think, 

 have said of his denominators b, c, d, &c. in § 368, " arbitrio nostro rehnquuntur, and 

 have thus left his readers to conclude, that the values adopted by htm were preferred 

 only on account of their convenience—" ita autem cos assunu convcnit —anil that a 

 different solution from his might be obtained by making a diflerent choice of dcnouima- 

 tors ; a conclusion which would be quite erroneous. 



JVet/;»S'mcj, VOL. XI. 2 e 



