the Value of a slowly converging Series. 185 
obtaining the value of this series, which have occurred to me, 
the easiest is that which I am now to describe, by which the 
business is reduced to the summation of two, three, or more 
series of this form, viz. ax — bx*-{- cx 3 — dx\ &c. and one 
•series of this form, viz. px n -\~ qx*' l -{- rx %n &e. where n is 
= 4, 8, 16, 32, or some higher power of 2. The investigation 
of this method is as follows. 
2. The series ax - f- bx x -\~ ex' -j- dx* -j- ex^fx 6 -^ &c, is 
evidently equal to the surnof these two series, viz. 
ax — bx^-\- ex 3 — dx*-\- ex s — fx 9 ,&c. 
* -f * 4- 2 dx* * + 2/.Z 6 , &c. 
of which, the value of the former is easily attainable, by the 
method so clearly explained, and fully illustrated, by Mr. Baron 
Maseres, in the Philosophical Transactions for the year 1777 ; 
and the latter, although it be of the same form with the series 
first proposed, yet has a great advantage over it, since it .con- 
verges twice as fast. Upon this principle, then, we may pro- 
ceed to resolve the series 2 bx*-\- zdx*-\- vfx e -\- zhx'-f zkx 1 * 
-}- 2 mx t% -\- &c. into the two following: 
9 .bx*™‘ <zdx*-\- ofx 6 — 2,hx 8 -\- zkx 10 — 2 mx 1% , &c. 
* -f 4 dx* * ~j~ & c - 
where, again, the value of the one may easily be computed ; 
and the other, although it be of the same form with the series 
at first proposed, yet converges four 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, 64, &c. times as fast, and consequently a few terms of it 
will be all that are requisite. 
An example, to illustrate this method, may be proper, 
which therefore is here subjoined. 
MDCCXCVIJJ. B b 
