Example

Tag 0BTT, permanent. Cite it as [Bourbaki, Tag 0BTT].

General Topology, chapter IV, § 7. Infinite sums and products of real numbers

If we take $u_{2n-1} = -1/n$ and $u_{2n} = 1/n$ for $n \geq 2$ , the product $(1 + u_n)$ is not absolutely convergent, since the series $(u_n)$ is not absolutely convergent; but since

\prod_{p=3}^{n} (1 + u_p) = \prod_{p=2}^{n} \left(1 - \frac{1}{p^2}\right),

and

\prod_{p=3}^{2n+1} (1 + u_p) = \left(1 - \frac{1}{n+1}\right) \prod_{p=2}^{n} \left(1 - \frac{1}{p^2}\right),

it follows from Theorem 4 that the product is convergent and that its value is

\prod_{n=2}^{\infty} \left(1 - \frac{1}{n^2}\right).

Moreover, it should be observed that the convergence of the series whose general term is $u_n$ is neither necessary nor sufficient for the convergence of the product whose general factor is $1 + u_n$ (see Exercises 21 and 22).

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

Cited by

Nothing.

Cites

Theorem 4