Example

Tag 08G6, permanent. Cite it as [Bourbaki, Tag 08G6].

Functions of a Real Variable, chapter V, § 4. APPLICATION TO SERIES WITH POSITIVE TERMS

Let $g(n) = \frac{1}{n}$ ; then

s_n = \sum_{m=1}^n \frac{1}{m} \sim \int_1^n \frac{dt}{t} = \log n

then

\frac{1}{n} - (\log n - \log(n-1)) \sim -\frac{1}{2n^2}

whence

s_n = \log n + \gamma + \frac{1}{2n} + o\left(\frac{1}{n}\right).

The constant

\gamma

which appears in this formula plays an important rôle in Analysis (cf. chap. VI and VII); it is known as Euler's constant; one has

\gamma = 0.577\,215\,664\ldots

to within

1/10^9

We shall see in VI, p. 288, how the Euler-Maclaurin summation formula gives an asymptotic expansion of arbitrary order for $s_n$ (or for $r_n$ ) in the most important cases.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.