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

PERFECT AND IMPERFECT SERIES.

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for all values of., except of course the value zero: but
the series being perfect, the limiting value of -, when (v
approaches 0 is zero; hence, by Lemma VI, Cor. 1, zero
must be the limiting value of
A0 + Ax + 4A,X2... AuV
when x approaches 0: now if Ao be not zero,
Ao + JA... AxnA AO
0 + A.. A becomes or o,
ogn 0
when x =0, and its limiting value cannot be zero when M
approaches zero; therefore A, = 0, and we have
Ao + AI... Ax" A, + A,2x... Anx"-l
tn X -1 1
except of course when x actually = 0.
Hence zero must be the limiting value of this latter quantity when x approaches o, which as before cannot be true
unless A = 0: and so we may go on and prove that A, = 0,
A4 0o...; and lastly, that
A,, + A~... Ani
X n
the limiting value of which cannot be zero unless An = 0.
Hence the principle of indeterminate coefficients holds for
a perfect series.
But the principle of indeterminate coefficients does not
hold for an imperfect developement.
For by what we have just proved, it is evident that if
A0, A,, A,... A, be each zero, then the limiting value of
R
when x approaches zero must be zero, and therefore the
series must be perfect. If therefore the series be not perfect,
A0, A,... An cannot be each zero, i. e. the principle of in.
determinate coefficients does not hold.
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