SERIES. 
A, aud the ratio r, then the sura is S = 
— — — . See Progression. 
r — 1 
Now, in a decreasing series from I, the 
more terms we actually raise, the last of 
them, A, becomes the lesser^ and the lesser 
A is rl — A is the greater, and so also is 
rl — A 
£ - : but rl — A being still less than 
rl, therefore ~ — ^ is still less than , 
r — 1 r — 1 
that is, the sum of any assignable number 
of terras of tlie series is still less' than the 
tI 
quote mentioned, W’hich is , and this 
r — 1 
is the first part of the theorem. 
Again, the series- may be actually con- 
tinued so far, tliat — — ^ shall want of 
' r — 1 
T X 
~ ^ less than any assignable difference ; 
^for, as the series goes on, A becomes less 
and less in a certain ratio, and so the series 
may be actually continued till A becomes 
less than any assignable number, (by Theo- 
rem 2 ) now — l J:~ ^ — and 
r — 4. r — 1 r — l’ 
A 
y - __ ^ is less than A ; therefore let any num- 
ber assigned be called N, we cto carry the 
series so far till the last term A be less than 
N j and because — wants of — 
r — 1 r — l’ 
A 
tlie difference — — , which is less than A, 
which is also less than N, therefore the 
second part of the theorem is also true, and 
rl 
■ is the ti ue value of the series. 
rl 
Scholium. The sense in which 
r — 1 
called the sum of the series, has been suffi- 
ciently explained; to which, however, we 
shall add this, that whatever consequences 
rl 
follow from the supposition of ^ being 
the true and adequate value of the series 
taken in all its infinite capacity, as if the 
whole were actually determined and added 
together, can never be the occasion of any 
assignable error in any operation or demon- 
stration where it is used fai that sense ; be- 
cause if it is said that it exceeds that 
adequate value, yet it is demonstrated that 
this excess must be less than any assignable 
difference, which is in effect no difference, 
and so the consequent error will be in effect 
no error ; for if any error can happen from 
rl 
- being greater than it ought to be, to 
represent the complete value of the infinite 
series, that error depends upon the excess of 
rl 
^ over that complete value; but this 
excess being unassignable, that consequent 
error must be so too ;- because still the less 
the excess is, the less will the error be that 
depends upon it. And for this reason we 
rl 
may justly enough look upon as ex- 
r — 1 
pressing the adequate value of the infinite 
series. But we are furthci- satisfied of the 
reasonableness of this, by finding, in fact, 
that a finite quantity does actually convert 
into an infinite series, which happens in the 
c^e of infinite decimals. For example, 
1 1 = . 6 6 6 6, &c. which is plainly a geo- 
metrical series from — in the continual 
ratio of 10 to 1 ; for it is 4- 4- - ^ 
10 ^ 100 ~ 1000 
+ 10000 ’ 
And reversely, if we take this series, and 
fiiid its sum by the preceding theorem, it 
comes to the same | ; for Z = -^ » — i o 
therefore r Z = ^ = 6 ; 
10 ’ 
10 ’ 
and r — 1 z= 9 j 
whence 
rl 
__ _6 _ 2 
r — 1 — 9 
We have added here a table of all the 
varieties of determined problems of infinite, 
decreasing, geometrical progressions, which 
all depend upon these three things, viz. the 
greatest term Z, the ratio r, and the sura S ; 
by any two of which the remaining one 
may be found: to which we have added 
some other problems, wherein S — L is 
considered as a thing distinct by itself; 
that is, without considering S and L sepa- 
rately. 
