Bourbaki

Exercise 11, § 4

Functions of a Real Variable, V, p. 51

Bourbaki marks this one as harder than the rest.

For every number a>0a > 0, prove the formula

(1!)−a/n+(2!)−a/n+⋯+(n!)−a/n∼1alog⁡n (1!)^{-a/n} + (2!)^{-a/n} + \cdots + (n!)^{-a/n} \sim \frac{1}{a \log n}
when nn tends to +∞+\infty (compare each term (p!)−a/n(p!)^{-a/n} with n−ap/nn^{-a p/n}).

Solution

No solution has been written yet.

Machine translation of the English, by gpt-5-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.