SERIES. 
notion of the infinity of a series, that is, 
of the infinity of the number' of its terras, 
as it is expressed in tlie definition. 
Hence it is plain th'at we cannot apply 
to an infinite series the common notion of a 
sura, viz. a collection of several particular 
numbers that are joined and added togedier 
one after another, for this supposes that 
these particulars are all known and deter- 
mined; whereas the terms <jf an mfinite 
series cannot be all separately assigned, 
there being no end in the numeration of its 
parts, and therefore it can have no sum in 
sense. But again, if we consider that the 
idea of an infinite series consists of two 
parts, ®i 2 . the idea of something positive 
and determined, in so. far as we conceive 
the series to be actually carried on; and 
the idea of an inexh-austible remainder still 
behind, or an endless addition of terms that 
can be made to it one after another, which 
is as different from the idea of a finite series 
as two things can be : hence we may con- 
ceive it as a whole of its own kind, which, 
therefore, may be said to have a total value 
whether that be determinable or not. Now 
in some infinite series this value is finite or 
limited ; that is, a number is assignable be- 
yond which the sum of no assignable num- 
ber of terms of the series can ever- reach, 
nor indeed ever be equal to it, yet it may 
approach to it in such a manner as to want 
less than any assignable difference; and 
this we may call the value or sum of the 
series; not as being a number found by the 
common method of addition, but as being 
such a limitation of the value of the Series, 
taken in all its infinite capacity, that if it 
were possible to add them all one after 
another, the sum would be equal to this 
number. 
Again, in other series the value has no 
limitation; and we may express this, by 
saying, the sum of the series is infinitely 
great; which, indeed, signifies no more than 
that it has no determinate and assignable 
value ; and, that the series may be carried 
such a length as its sum, so far, shall be 
greater than any given number. In short, 
in the first case, we aflarm there is a sum, 
yet not a sum taken in the common senses 
in the other case, we plainly deny a deter- 
minate sum in any sense. 
Theorem 1. In an infinite series of num- 
bers, increasing by an equal diffei ence or 
ratio (that is, an arithmetical or geometrical 
increasing progression) from a given num- 
ber, a term may be found greater than any 
assignable number. 
Hence, if the series increase by differences 
that continually increase, or l)y ratios that 
continually increase, comparing each term 
to the preceding, it is manifest that the 
same thing must be true, as if the differences 
or ratios continued equal. 
Theorem 2. In a series decreasing injn- 
finitum in a given ratio, we can find a term 
less than any assignable fraction. 
Hence, rf the terms decrease, so as the 
ratios of each term to the preceding do 
also continually decrease, then the same 
thing is also true, as when they continue 
equal. 
Theorem 3. The sura of an infinite series 
of numbers all equal, or increasing contum- 
ally, by whatever differences or ratios, is in- 
finitely great ; that is, such a series has no 
determinate sum, but grows so as to exceed 
any assignable number. 
Demons. First, if the terms are all equal, 
as A : A : A, &c. then the ^um of any finite 
number of them is the product of A by 
that number, ai An; but the greater n is, 
the greater is An; and we can take ?i 
greater than any assignable number, there- 
fore A n will be still greater tlran any assign- 
able number. 
.Secondly, suppose the series increases 
contuiually, (whether it do so infinitely or 
limitedly) then its sum must be infinitely 
great, because it would be so if the terms 
continued all equal, and therefore w ill be 
more so, since they increase. But if we 
suppose the series increases infinitely, either 
by equal ratios or differences, or by increas- 
ing differences or ratios of each term to the. 
preceding ; then the reason of the sums be- 
ing infinite will appear from the first theo- 
rem ; for hi such a series, a term can be 
found greater than any assignable number, 
and much more therefore the sum of that 
and all the preceding. 
Theorem 4. The sum of air infinite series 
of numbers decreasing in the same ratio is 
a finite number ; equal to the quote arising 
from the division of the product of the ratio 
and first terra, by the ratio less by unity ; 
that is, the sum of an assignable number of 
terms of the series can ever be equal to 
that quote ; and yet no number less than it 
is equal to tlie value of the series, or to 
what we can actually determine in it; so 
that we can carry the series so far, that the 
sum shall want of this quote less than any 
as.siguable difference. 
Demons. To whatever assigned number 
of terms the series is carried, it is so far fi- 
nite ; and if the greatest term is I, the least 
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