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

82

TAYLOR S SERIES.

Hence I think the importance of distinguishing series into
perfect and imperfect is manifest.
A function 133. Lastly, we may remark, that if f(x) be a function
which has
only one which has only one value for each value of x, it cannot be
value for
each value developed in more than one perfect series.
of x can be
developed For if possible let the two different perfect series
feetin only peries.
feseries. + A + A,2...A4An +, and Bo + B1x +B2x2...B,,iv + S,
be both equal to f(x), then sincef(m) has only one value for
each value of x, these series must be equal to each other; and
therefore we have
A- Bo + (Al- B,) + (A - B,) x2..+ (A- B,) + i~ - S = o.
Now when x approaches o, the limiting values of R and of
S R-S
- are zero, therefore so also is that of: hence, by what
has been proved in the preceding Article, we must have
Ao = Bo, A1 = B, A2 = B2... A, = B,,, and therefore R = S;
hence the two series are identical; and therefore f(x) cannot
be developed in two different perfect series.
Having made these preliminary remarks, we now proceed
to shew the manner in which the Differential Calculus may be
applied to the developement of functions in perfect series.
Proposi- 134. To develope f (a + h) in a perfect series of powers
tion.
To de- of h.
velope
f(a+ih) in
a perfect In Lemma XXI. put a + h for x, and we have
series of
powersofh. h h. hn
f(a + ) =f (a) f'(a) +f2(a) -...(a) + Qn (1),
here Q - is the remainder; call it R, then n = -; now
Fn h'1 f n o
zero is the limiting value of Q when x approaches a, i. e. when
h approaches zero, (see Lemma XXI.), and therefore the same
is true of -; hence (1) is a perfect developement. Since
hn