Example 1

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

Functions of a Real Variable, chapter V, § 3. ASYMPTOTIC EXPANSIONS OF FUNCTIONS OF A REAL VARIABLE

Let $f(x) = 1/\log x$ ( $x > 1$ ); we have seen that $\int_a^\lambda dt / \log t \sim x / \log x$ , and that the difference $\int_a^\lambda \frac{dt}{\log t} - \frac{\lambda}{\log \lambda}$ is a primitive of $1/(\log x)^2$ ; one can apply prop. 8 again to this function, so obtaining $\int_a^\lambda dt / (\log t)^2 \sim x / (\log x)^2$ . By recursion one thus obtains the expansion

\int_a^\lambda \frac{dt}{\log t} = \frac{x}{\log x} + \frac{x}{(\log x)^2} + \frac{2x}{(\log x)^3} + \cdots + (n-1)! \frac{x}{(\log x)^n} + o \left( \frac{x}{(\log x)^n} \right).

Note that, irrespective of $n$ , all the terms of this expansion tend to $+\infty$ with $x$ .

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

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