| (b − a)n | (1 − θ)n−1 ƒn {a + θ(b − a) }. |
| (n − 1)! |
The conditions of validity of Taylor’s expansion in an infinite series have been investigated very completely by A. Pringsheim (Math. Ann. Bd. xliv., 1894). It is not sufficient that the function and all its differential coefficients should be finite at x = a; there must be a neighbourhood of a within which Cauchy’s form of the remainder tends to zero as n increases (cf. Function).
An example of the necessity of this condition is afforded by the function f(x) which is given by the equation
| ƒ(x) = | 1 | + Σn=∞n=1 | (−1)n | 1 | . | |
| 1 + x2 | n! | 1 + 32n x2 |
(i.)
The sum of the series
| ƒ(0) + xƒ′(0) + | x2 | ƒ″(0)+ ... |
| 2! |
(ii.)
is the same as that of the series
e−1 − x2 e−32 + x4 e−34 − ...