VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 313 



series of this form, viz. ax — bx 2 + ex 3 — dx\ &c. and one series of this form^ 

 viz. px n -j- qx 2n -\- rx 3 " -j- &c. where n is = 4, 8, ] 6, 32, or some higher power of 2. 

 The investigation of this method is as follows. 



2. The series ax -f- bx 2 -J- ex 3 + dx 4 -\- ex 5 + /^ 6 + & c « i s evidently equal to 

 the sum of these two series, viz. 



ax — bx 2 + ex 3 — eta 4 + ex 5 — fx 6 , &c. 



* + 2bx 2 * -f 2<&? 4 * + 2fx 6 , &c. 

 of which, the value of the former is easily attainable, by the method so clearly ex- 

 plained, and fully illustrated, by Mr. Baron Maseres, in the Philos. Trans, for the 

 year 1777 ; and the latter, though it be of the same form with the series first pro- 

 posed, yet has a great advantage over it, since it converges twice as fast. On this 

 principle then we may proceed to resolve the series 2bx 2 -f- 2dx 4 -f- 2fx 6 + Q,kx 9 

 -f 2kx 10 -f 2mx 12 -L. &c. into the two following : 



2bx 2 — 2dx 4 + Qfx 6 — 2hx 8 -f 2kx 10 — 2mx 12 , &c. 

 * + 4dx 4 * + 4hx s * + 4mx 12 , &c. 



where again the value of the one may easily be computed ; and the other, though 

 it be of the same form with the series at first proposed, yet converges 4 times as 

 fast. And in this manner we may go on, till we obtain a series of the same form 

 with the series at first proposed, which shall converge 8, 16, 32, 6*4, &c. times as 

 fast, and consequently a few terms of it will be all that are requisite. 

 An example, to illustrate this method, is here subjoined. 



3. Let it be proposed to find the value of the series x -\- — -\- —- -\- -\ (- 



* ft , . q 



-~- + &c. ad infinitum, when x = — . 

 o 10 



4. In order to obtain the sum of this series, with the less work, it will be requisite 

 to compute a few of the initial terms, as they stand. For, if we begin the opera- 



r 2 r 3 _4 



tion with computing the value of # 2~ + T """ T* ^ c * ^ tbe differential series 



before mentioned,* the values of d', d /; , d'", &c. will be -, ^-, y^, &c. respec- 

 tively, i. e. -£-, •£-, -£-, &c. which is a series decreasing so very slowly, that the only 

 advantage obtained by this transformation of the series is in the convergency of the 

 powers of instead of the powers of x, which indeed is very great ; for, x 



bein S = lo' TTT is m f§ ; so that the new ■*** J§ + * ' (^> 2 + * • (|) 3 + 



* • (^) 4 + &c « thou g h = J5 ~ * ' (£)* + i • (^) 3 - * • (^,) 4 ' &c « y et converges 

 more than 7 times as swiftly. But, if we begin the work by computing the 



* The theorem best adapted to this business is the following ; viz. ax — bx* + cx s — dx*, &c. = 

 ax d' x'' b" x i d'" x* 



X+T + JTTxr + UTW + JTTxy + &c ' D being = a ~ b > D - * ,n . *A e > £ 



= a — 3b + 3c — d, &c. See Scriptores Logarithmici, vol. 3, p. 290, where b, c, d, &c. denote the 

 •ame quantities that a, b, c, &c. do here. 



VOL. XVIII. S S 



