Remark

Tag 088O, permanent. Cite it as [Bourbaki, Tag 088O].

Functions of a Real Variable, chapter II, § 2. INTEGRALS OVER NON-COMPACT INTERVALS

An integral can be convergent without being absolutely convergent; this is what is shown by Example 3 of II, p. 64, where the series with general term $u_n$ is convergent without being absolutely convergent.

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

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