Exercise 13, Appendix 0

Functions of a Real Variable, V, 266

Let $f$ be an (H) function such that $1 \ll f(x) \ll x$ . Show that in the Taylor formula

f(x + f(x)) = f(x) + f(x)f'(x) + \frac{1}{2!}(f(x))^2 f''(x) + \cdots +

+ \frac{1}{n!}(f(x))^n f^{(n)}(x) + \frac{1}{(n+1)!}(f(x))^{n+1} f^{(n+1)}(x + \theta f(x))

with

0 < \theta < 1

(I, p. 42, exerc. 9) each term is negligible relative to the previous one. If

f

is of order

< 1

relative to

x

, the last term in this sum tends to 0 with

1/x

once

n

is sufficiently large.

Solution

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