An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 27. 28. Algebraic and exponential functions. 51
Y is positively arbitrarily small when either a > 1 and Ax > 0,
or a < 1 and Ax < 0, in other cases negatively arbitrarily small.
1 1 __
Writing -= m the expression becomes — 1 Now
o 1 +
we have to show, that (1+ i l) has a determinate finite limiting
value, when mn passes through the continuous series of numbers or
through any discontinuous series of numbers whose limiting value
exceeds any finite amount. At the same time we have to conduct the
investigation in a way which shall present a convenient method of
calculating this value.
Let us first make mn pass through the series of positive integers,
then by the binomial theorem as just proved, we have always:
(1+ i)n 1F j + n (I ( 1 ) ) ( 1 )2
- 1 __1 + 1 +, (1 - l-) 1. (1 - A) (1 p- 2); +... +1+I. 2 M M-3 1_-31_
= 22 + R.
Where by Z, we signify the sum of the first n terms, by R the sum of
the last terms reckoned on from the (n + 1), so that therefore:
= (1 — ) (1 -). *.. (1 - -r ) - * S
S embracing the m + 1 - n terms
S=I (1-I-) 1 + ( (l IZ) (1 +) 1
n- + (1 it), * + ~c t + (n -+ 2) +- ~
+- (\ l - 1 1-
1 \ m V mn (n-L 1) '' *n
If m be'a considerable number, we can approximate arbitrarily to the
value of (1 + -)n by merely summing the first n terms of the series,
n being a number much smaller than m.
1 2
For, the differences 1 -, 1- 2 etc. in the expressions for B
and S being positive proper fractions, the remainder R is certainly
smaller than the value obtained when we put in it unity for each of
these differences; and a fortiori, < I 1 + -+1+ -i + i+ + (-, Ir-) -
-( - I A, — -
11
1 I t +1
still more is: R < 1 1: n '
1 + —4
4*