SERIES. 
ner, by dividing 1 by the algebraic sum 
or by a—c, the quotient will be in 
these two cases, as below, viz. 
1 1_ 
a-]-c a a'*’ 
1 '_1 I c I I 
a — c C’ 
where the terms of each series are the same, 
and they diflfer only in this, that the signs 
are alternately positive and negative in the 
former, but all positive in the latter. 
And hence, by expounding a and c by 
any numbers whatever, we obtain an end- 
less variety of infinite series, whose sums or 
values are known. So, by taking a or c 
equal to 1, or 2, or 3, or 4, &c. we obtain 
these series, and their values ; 
^ ^ =1— 1-f 1 — 1 + 1 — l,&c. 
1 + ^ 
. i 1 
O ^ .52 T Q 3 1 
2 + 1 
1 
3’ 
.i+1. 
22 T 23 
. — -= 1—2 + 2 ^ 
1 + 2 3 ‘ 
1 -1 1 I ^ 
3 + i~4— 3 
, &c. 
-2^&c, 
1 
32-1-33 34' 
&c. 
And hence it appears, that the same 
quantity or radix may be expressed by a 
great variety of infinite series, or that many 
different series may have the same radix, or 
sum. 
Another way in which an infinite series 
arises, is by the extraction of roots. Thus, 
by extracting the square root of the number 
3 in the common way, we obtain its value 
in a series as follows, viz. 1-73205, 
„ _1 iltAj g I 5 
&c. ; in w'hich w'ay of resolution the law of 
the progression of the series is not visible, 
as it is when found by division. And 
the square root of the igebraic quantity 
((2 + gives 
v/us + c' = a+ — — ^ +r:^,&c. 
^ ‘ ‘2a 8 a^ ' 16 a’’ 
And a 3d way is by Newton’s binomial 
theorem, which is an universal method, that 
serves for all sorts of quantities, whether 
fractional or radical ones : and by this 
means the same root of the last given quan- 
Hence it appears that the signs of the 
terms may be either all plus, or alternately 
plus and minus, though they may be varied 
in many other ways. It also appears that 
the terms may be either continually smaller 
and smaller, or larger and larger, or else all 
equal. In the first case, therefore, the se- 
ries is said to be n decreasing one; in the 2d 
case, an increasing one ; and in the 3d case, 
an equal one. Also the first series is called 
a converging one, because that by collecting 
its terms successively, taking in always one 
terra more, the successive sums approxi- 
mate or converge to the value or sum of the 
whole infinite series. So, in the series 
-^ = i=i + i+ i + ^ &c 
3 — 1 2 3^9^27^81’ 
’ 1 . 1 
the first term ~ is too little, or below 
which is the value or sum of the whole infi- 
nite series proposed ; the sum of tlie first 
two, terms i + i is ^ = -4444, &c. is also 
too little, but nearer to ~ or -5 than the for- 
mer; and the sum of three terms - + i + 
3 ^9 ~ 
1 . 13 
^ >s — = -481481, &c. is nearer than the 
last, but still too little ; and the sum of four 
terms 
3 + 9 + 27 + 8i in = •49-’827', &c. 
which is again nearer than the former, but 
still too little; which is always the case 
when the terms are all positive. But when 
the converging series has its terms alter-- 
nately positive and negative, then the suc- 
cessive sums are alternately too great and 
too little, though still approaching nearer 
and nearer to the final sum or value. Thus, 
in the series 
+1 4 3 9 "^27' 81’**^’ 
the 1st term | = -333, &c. is too great ; 
two terms -222, &c. are too little; 
three terms i — i + 1 = -259259, &c. 
are too great; 
four terms- ■ 
. 1 + 1 .. 
9 ~27 
'81 
-24691S 
&c. are too great, and so on, alternately, 
too great and too small, but every succeed- 
ing sum still nearer than the former, or 
converging. 
In the second case, or when the terras 
