^ Mr Herscliel on the applicatmi (^a new mode of Analysis 
(4.) Let us now consider the series 
S r= Ao- 0” + A,, r -h A,, a” 4- &C. 
By substituting here for O**, 1^, S”, &c. their values derived 
from the equation 
(1 + A)”^, = 
we find 
S = Ao- (1 -f A)«0” q- Aj. (1 4- A) 0” -f A, (1 4- A)2 o'" 4- &c. 
or, separating the symbols of operation from those of qurmtity, 
S= {a„ a + A)»+A, (1+A) + A, (1 + a)^ + &c.|o’‘ 
= F (1 + A) O”- 
if we suppose that the co-officients A^, Aj, &c. are such that 
F {{) — Aq 4-Aj ^q-A^ 4- 
We will exemplify this in a few particular instances. 
(5.) Suppose we begin with the series 
S = 4- — &c. acl inf 
Here we have 
r(o = <-i« + &c.=(~ 
and therefore 
l+Ao^. 
4- A 
This, then, is the abbreviated expression for the sum of the pro- 
posed series, but if we would obtain its numerical value, we 
must reduce this to a limited number of calculable terms, by de- 
veloping 
1 4- A 
2 4- A 
- in 
powers of A, and rejecting all beyond the n^^. 
thus we get 
^ 0^ aO” 
2 4 
+A’‘0” 
8 4 1 
and if we assign to n particular numerical values, we obtain 
precisely the same results with those long ago found by Euler. 
If we take the series 
I'lH" S” ^^4- &c. 
which is one of very frequent occurrence in analysis, our for- 
mula gives for its sum to ,x terms 
<(1 + A) — + Vl + A)^'*' V 
1 — iS(l + A) 
