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Exercise 11, § 4

Functions of a Real Variable, V, 263

For every number $\alpha > 0$ derive the formula

(1!)^{-\alpha/n} + (2!)^{-\alpha/n} + \cdots + (n!)^{-\alpha/n} \sim \frac{1}{\alpha} \frac{n}{\log n}

as

n

tends to

+\infty

(compare each term

(p!)^{-\alpha/n}

to

n^{-\alpha p/n}

).

Solution

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