Exercise 5, § 2

Functions of a Real Variable, VII, 328

Let $x_n$ be the root of the equation $\Gamma'(x) = 0$ in the interval $]-n, -n+1[$ . Show that

x_n = -n + \frac{1}{\log n} + O \left( \frac{1}{(\log n)^2} \right)

(use the complements' relation and formula (5) of VII, p. 316). Deduce that

\Gamma(x_n) \sim \frac{(-1)^n}{\sqrt{2\pi}} \ n^{-n-1/2} \ e^{n+1} \ \log n .

Solution

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