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.

~ 38-41.

Taylor's Theorem.

69

the remainder of the series after n places, we can formulate
the condition of convergence also as follows:
The necessary and sufficient condition for the convergence of an
infinite series consists in this, that for any number d however small,
a place n can be found in it, such that its remainders Rn, vRn+, 1 R-+2,..
are always smaller in amount than 4.
This cannot possibly be fulfilled unless the amounts of the terms
in the infinite series ultimately decrease and have zero as limit, but
this condition alone is not sufficient for its convergence. The value
of an infinite series which does not converge, will either be quite
indeterminate, when the series of sums Sn oscillates between arbitrary
values, or it will be determinately infinite positively or negatively.
In both cases the series is said to be divergent.
40. Now the forms developed in ~ 37 express the simple functions
as infinite series of powers. For, supposing the value of the
function and of all its derived functions be known for the argument a,
and that the value of the function for any other x is required, then
in consequence of these equations we have, putting x for b:
f (x) = f (a) + (X - ") f (a) + (x ) f" ( +.. (X:-alf n-1 (a)+ R
R (x- a) (a + 0 (x -ao)), or (x- a)n-O) (a + (x —a)).
On the right side accordingly all terms are known, except the last, in
which the unknown fraction 0 occurs. But if we can prove that this
last term B, formed for arbitrarily increasing values of n, passes
through a series of numbers having zero as limit, then neglecting
this last term, we shall obtain the value of f (x) with arbitrary approximation by summing as many terms as we please of the infinite series:
f(x) -- f(a) + (x -aC) f'(a) + (x- a)f(a) (x-a) f"' (a) +. in c.
f2 3) = ffa ~ ( - a) (-f'
This is Taylor's series named after its discoverer*); it teaches: If we
Inow the value of a function and of a ll its derived functions for a single
argument, we can calculate the value of the function for every other
argoument x C a, i in the interval from a to x the fttnction and as
many of its derived ft6nctions as may be formed are continuous without
becoming infinite, and if Limr (B) vanishes for n = oo.
41. The examination of the first hypothesis is apparently complicated, requiring the finiteness and continuity of all the derived
~) Taylor (1685-1731) in his chief work: Methodus incrementorum directa
et inversa, 1715, established this series but without taking account of the remainder
term. Mac Laurin (1698-1746) in his Treatise of Fluxions, 1742, developed the
series for the special value a = 0.