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.

~ 45-46. Binomial series. 79
46. Binomial series.
yJ-f(x)-(1l - x). 1. n having any value; y always positive.
For x -- 0 the value of y and of all derived functions is known; we have
f"(x) = (m - 1) (nm - 2) (m -- n + 1) (1 -- x)n-^.
These derived functions are continuous as long as x > - 1; therefore
x 11v( 1) 2 m ( - 1) ( (m -- 2)
+( - x)' = 1 + x+ 2- x + * *. (1s - 1) (- 2) *.. (m- + - 2) _1 +;2
=_ vn, xn (1 + Ox)'"- or - 1 n nx 1 m (1 - )n-1 (1 + 0X)' —";
1___ (mb - 1) (m - 2).. (m - i + 1)
It is convenient to consider the second form of the remainder. Since
x > - 1, 1 +- Ox is a positive number for all values of 0 as required.
Let us put
(I - + x)
(}^02- -, (1+OX)'-~.^Xc —,
then we have
in - v -2 -- 2 v -k t - ( —t - -)
1 1 2 - 1 -
(mv-1)Z (m -2) (m- -k)z (m-(n-1)) E
1 2 7k n-1
The factors E and m are finite. The product will certainly have its
limit zero, when its factors begin somewhere to be proper fractions and
remain proper fractions when n becomes oo. For, if G be numerically
the greatest of the fractions between (2m — 7) and m ~ -( -), the
product of these factors taken absolutely is less than G"-k; but such
a power has zero as limit. On the other hand the product will certainly increase beyond all linits, provided the factors once become
greater than unity and remain so.
But now since as n increases, the amount of - ) 1 approaches
i- 1 i a pproahes
arbitrarily to that of z, the amount of z must be less thlan unity;
therefore
for x>O, (1 - ) x < o or(l - ) x < 1 - Ox, i.e.x 1,
1 +ox<1
forx<(O (1 - O) x
forl+ x > - or (1 - ) x > -1 - x, i.e. x>-l.
Result: lf - 1 < x < + 1, the positive function (1 + x)"Y can
be calculated for every mn with arbitrary approximation from the
infinite sum: