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.

66 Calculation of functions by infinite series. Bk. I. cb. VIIT.
K /..' (x )
3
so that
mf() == f ((t) + (b -w a)f'(a) + ( —T f",, (a) + (b 3- a f,,, (a, + O (b -- c,).
We introduced the quantity I2 in the denominator from the first, in
order that the equation for K arising from differentiation might assume
as simple a form as possible.
Let us now put somewhat more generally for any value of n:
/[b) — f(a)+-(b — a)fa'() a) f-I a) '~. n —1l(a) -(b — ai,
2 InI f P
where p is to signify any positive integer, and let us enquire whether
K can be expressed by values of the tth derived. Once more, the
function
p(x) = f(b) - f(x) - (b - x) f'(x) - (7 -f''X)f(x) -
b... _ xn-1 (f1() -- )P
is continuous, everywhere finite, has a determinate differential quotient,
and vanishes for x = a and for x =- b: so that we must have
() - {(b - X (b - - 0
s (xim) be In-i_ (x1) (b x)P-1 K = O
or, as x1 must be different from b:
(b - X,)n - (b - ca)n-p(1O)n f+p
K (b-,) n- fn() (X) + (b - a))
Accordingly
(b - (b' a
(b) = f() (b - a)f'(a) + ( a2 f(a)+. - fn — (f )
+ (b-a) (1- O)p (a + 0 (b -- a)).
Thie last term assumes particularly simple forms when p is put
or, or = 1, we have
I f(b) == f(a) + (b - a) f (a) + b- f" (a) + - n '- f-1(a)
L I —I f
+ (n -)/ (a( + 0 (b - a)), or
II. f(b) = f(a) + (b -- a) f'(a) + ( * a (a) (b -- / n —
(b - a) 7' 0) n f (
+ ( -a)( - ) f(c + (b - a)).
0 does not signify the same value in both equations, moreover all