i86 Mr. Hellins's new Method of computing 
3. Let it be proposed to find the value of the series j; + — 
+ Y + + + + & c - a d infinitum, when x = 
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 operation with computing 
the value of x — — A , &c. by the differential series be- 
2 1 3 4 J 
fore mentioned *, the values of D', D", D 7// , &*c. will be — , 
2 2.3 
&c. respectively, i. e. t, &c. which is a series de- 
creasing 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 
I ■ J" X 
very great ; for, x being = is = ; so that the new 
series * + i • + f-ftT ■ + i . Sf-+ although = - 
\ &c. yet converges more than seven 
times as swiftly •f. But, if we begin the work by computing the 
first eight terms of the series, as they stand, and then compute 
the value of -L — -L -f“ 77 — 77 +? infinitum , by the same 
theorem, the values of D', D", D /7/ , &c. will be ■, 
10 11 1 i 3 & c ' 1S a ser i e S decreasing, for a great number 
* The theorem best adapted to this business is the following; viz. ax — bx % - f- 
D" .r 3 D® or 4 
, ... ox D' x % 
cx 3 — dx*, &c. — — h 
+ 
l+X ‘ r (1-f.z) 3 + (I+^) 4 
— b, D"— a — 2 b+c, D'"— a — > 36 -f 3c — d, C2?c. 
See Scriptores Logaritbmici , Vol. III. p. 290, where b, c, d, &c. denote the same 
quantities that a, b, c, &c. do here. 
— \ 7 
f *2- is = 0.4736842, and— 1 is = 0.4782969. 
*19 1Q| 
4- &c. D' being — 
a 
