» Direct 
Method. 
“te 
——~ 
—dx  dzx Quadu 
as tall i + 
-FLUXIONS. 
pall ie pes jhe Sea: 
ie Cor an a Ca 
Zz te . 
eset tbe phe. 
y: ‘ 
| RES 
a(l+r) _ ah; 
ee Tach git 3723 4-42rt-+&e, 
S++ 2), porte giet pater p&e. 
(l—2)" &e, &e &e. 
By the principle employed in this article, we may 
discover as many series, finite or infinite, as we please, 
that may be summed, - z 
37. As an example of another mode of investigating 
series that may be summed, let us assume’the series of 
identical equations. 
1—2*=(1+42)(1—2), 
1—z'= bet et 
1—2*=(1 +2*)(1—2'), 
1214 )2): 
By te the ucts of the two sides, and leaving 
8 oe bi ee to both, we obtain 
l—x*"=(1l—z)(14+2)(142*)(142) ... *(1+42"). 
Now it et ee ay oo 29. : that EOE r, 
* t, v, &e. to be any funetions of «, if y=rstv, &e. 
en 
¥y 
Therefore, assuming that 
y=l—a™, r=1—a, s=1 44, t=1 4-2, v= 1424, &e. 
and taking the fluxions, we have 
4x3d x 
i-a 
fet Type 
nv dae 
¥en”? 
and hence, by dividing all the terms by dx, and multi- 
plying by «, and transposing the — we find 
x 2nw ; 
=< 1a F tg 
a 2x" a n 
=Tyatige Tite +143 
This elegant analytical theorem holds true, indepen- 
dently of any particular value of x ; and whatever be the 
‘number of the terms, observing that the exponents of 
1—c** 
the powers of x must be the terms of the 
series 1, 2, 4, 8, &c. the common ratio of which is 2. 
38. Let us next assume this. other series of identical 
equations. 
a—1=(e—1)(2? 41, 
- hota (2t —1)(24-41), 
eis(e—ayat 41), 
of the two series of equations, and i 
ae to both, we find «1 expressed 
407 
(ot —1) (24.1) @t41)(2* 41)... X @P41). 
Hence, paaies. as in last article, dividing the fluxion “~~ 
of each Factor by factor itself, and leaving out dz, 
which is common to all the terms, we find wis 
w—l1 
a 1 j-1 ries s 
x : 43 z +t 
n(a —1) x41 rt] 
at ent 
+3 zx = x] +4 L > 
vet oe oe | 
ee am and multiplying all the terms by z, 
we ‘ 
2" = + 
n(2#—1) ag: 
x an 
a +} + § 
ve 41 ety erp 
This equation holds true, whatever be the number of 
terms: but let’us now suppose their number infinite ; 
rt 
cn 
then the numerator of the expression becomes 
of x, 
n(c*—1 
=1, and the. denominator is the Napierean 
(Art. 12.) Hence we have 
be aren 
L(z) 2—1 
: : 
ole oo teh 
er at+l1 vel 
We may give this expression another form, by wri- 
ting ~ instead of x, and afterwards changing y into =, 
+'&e.) 
and observing that log. (=) =— log. (x); we then 
have. 
1 1 
T(z) “21 : 
1 1 
5 er pers 2 Soe po Free 
. av? 41 at4t at 41 
By adding the corresponding sides of these two for« 
ha we get this third form 
1 yttt 
Oi 
4. 4 
-(34 43-1 +45 —l +e.) 
a4 x4 zi4i 
which is better adapted to calculation than either of the 
others, because it converges faster ; but it does not con< 
bb! abating’ whi we have in Art. 11. 
y taking the fluxions of both sides of this equation, 
ws sar the ex cp Mah ali ay of the 
uare zr), W investiga a diffe- 
peg Art. 11; and repeating the pescess, we 
ma’ ‘or 
mer ewe he’ bp the reciprocal of its, third 
highes ers. 
39. By Me RarinukHie of Sines, formulz (G), 
Sin. z=sin. } 2 x 2 cos, 4.x, and sin, 3 a= 
sin, .¢X2 cos, 4x, 
‘ore, 
