Exercise 3, § 2

Functions of a Real Variable, III, 125

For every real $x > 0$ and every complex $m = \mu + i\nu$ one puts $x^m = e^{m \log x}$ ; show that the formula (19) of III, p. 108 remains valid for $m$ complex and $x > -1$ and that the remainder $r_n(x)$ in this formula satisfies the inequalities

|r_n(x)| \leq \left| \binom{m-1}{n} \frac{m}{\mu} x^n [(1+x)^{\mu} - 1] \right| \quad \text{if } \mu \neq 0 \\ |r_n(x)| \leq \left| \binom{m-1}{n} m x^n \log(1+x) \right| \quad \text{if } \mu = 0.

Generalize the study of the convergence of the binomial series to the case where

m

is complex.

Solution

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