An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

CONVERGENCY OF SERIES.

87

For then it is clear that
({ - a)r'
f n (V) -_ f n(a)} (x )
h" h" h1'
and therefore —, N-, and therefore Q - may be
Fn Tn I'n
diminished ad libitum by sufficiently increasing n: therefore
by (130) the developement is converging, and we may assert
that
h h2
f(a + h) f (a) + f (a) - + f (a)
h3
+ f3 (a).... ad infinitum.
j- 138. If, for all values of n greater than a certain value A simpler
condition of
Un +l converr, never exceeds a certain ratio a which is less than gencyr
r~ gency.
Un
unity, the developement is converging.
U ZC /Ur+3
For then we have +l not > a +2 not > a not
Ur Ur+ i Ur+ 2
> a.... --- not > a; and therefore, multiplying these in/n —1
equalities, we have - not > a-'', or u, not > an-"u.. Now
Ur
since a is less than unity, the limiting value of a"-r when
n approaches co is zero, and therefore, zero is also the limiting value of an-',., and therefore of tu,: consequently the
developement is converging.
139. By means of Lemma XXI. we arrive at the con- Taylor's
clusion, that there necessarily exists a perfect developement of rived atr
f(a + h) in the form somewhat
differently.
A0 + Ah + A2 h... + Ahn + R,
provided none of the quantities f(a), f(a), f2(a)... f(a)
be infinite: we shall now prove the same thing somewhat
differently.