SERIES. 
grow larger and larger, the series is called a 
^kierging oue, because that by collecting 
the terms continually, the successive sums 
diverge, or go alvi'ays further and further 
from the true value or radix of the series ; 
, being all too great when the terms are all 
positive, but alternately too great and too 
little when they are alternately positive and 
negative. Thus, in the series 
-^=1 =1 — 2+4— 8, &c. 
1+2 3 ^ 
the first term + 1 is too great ; 
two terms 1 — 2 = — 1 are too little ; 
three terms 1 — 2 + 4 = + 3 are too great ; 
four terms 1 — 2 + 4 — 8 = — 5 are too 
little; and so on, continually, after the 2d 
term, diverging more and more from the 
true value or radix i , but alternately too 
O 
great and too little, or positive and negative. 
But the alternate sums would be always 
more and more too great if the terms were 
all positive, and always too little if nega- 
gative. 
But in the third case, or when the terms 
are all equal, the series of equals, with al- 
ternate signs, is called a neutral one, be- 
cause the successive sums, found by a con- 
tinual collection of the terms, are always at- 
tire same distance from the true value or 
radix, but alternately positive and negative, 
or too great and too little. Thus, in the 
series 
— L = 1 =1 — 1 + 1 — 1 + 1 — &c. 
1+12 ‘ 
the first term 1 is too great ; 
two terms 1 — 1 = 0 are too little; 
three terms 1 — 1 + 1 .= 1 too great ; 
four terms 1—1 + 1 — 1=0 too little ; 
and so on, continually, the successive sums 
being alternately 1 and 0, which are equally 
different from the true value, or radix, i, 
the one as much above it as the other be- 
low it. 
A series may be terminated and render- 
ed finite, and accurately equal to the sum 
or value, by assuming the supplement, after 
any particular term, and combining it with 
the foregoing terms. So, in the series ^ — 
i-J-l— — , &c. which is equal to i, and 
4~8 16’ ^3’ 
found by dividing 1 by 2 + 1, after the 
first term, of the quotient, tlie remain- 
der is — 1, which, divided by 2 + 1, or 3, 
1 
gives — - for the supplement, which, com* 
1 . 1 i 
bined with the first term, -, gives - — g 
= the true sum of the series. Again, af- 
1 1 
ter the first two terms the remain- 
2 4’ 
der is + -, which, divided by the same di- 
4 
visor, 3, gives ~ for the supplement, and 
j 1 
this combined with those two terms - 
4 ‘12 4* 12 
4> 
4 1 
^ or - the 
makes , 
■ 2 4 ‘ 12 4 * 12 12 
same sum or value as before. And, in ge- 
neral, by dividing 1 by a + c, there is ob- 
tained 
a + c “ 
0” + * 
1 _ -i -L fl 
a ' a3 
a“ + ‘ (« + c) 
at any term, as 
c” + ‘ 
± ji+r + 
; where, stopping the division 
the remainder after 
this term is 
the same divisor, a + c, gives 
c” 
I 
which, being divided by 
c'>+‘ 
(a + c) 
for the supplement as above. 
“The Law of Continuation.” — A series 
being proposed, one of the chief questions 
concerning it is to find the law of its conti- 
nuation. Indeed, no universal rule can be 
given for this ; but it often happens, that 
the terms of the series, taken two and two, 
or three and three, or in greater numbers, 
have an obvious and simple relation, by 
which the series may be determined and 
produced indefinitely. Thus, if 1 be di- 
vided by 1 — X, the quotient will be a geo- 
metrical progression, viz, 1 + a; + + x\ 
&c. where the succeeding terms are pro- 
duced by the continual multiplication by x. 
In like manner, in other cases of division, 
other progressions are produced. 
But ill most cases, the relation of the 
terms of a series is not constant, as it is in 
those that arise by division. Yet their re- 
lation often varies according to a certain 
law, which is sometimes obvious on inspec- 
tion, and sometimes it is found by dividing 
the successive terms one by another, &c. 
Thus, in the series 
dividing the 2d term by the 1st, the 3d by 
the 2d, the 4th by the 3d, and so on, the 
quotients will be 
