Convergence of $\prod_{n\in\mathbb{Z}}(1+a_n)$ when $\{a_n\}\in\ell_1(\mathbb{Z})$

Show that $\prod_{n\in \ZZ} (1 + a_n) < \infty$ if $\ts{a_n} \in \ell_1(\ZZ)$.

Solution

1. Since $\{a_n\} \in \ell_1(\ZZ)$, we have $\sum_{n \in \ZZ} |a_n| < \infty$.

definition of $\ell_1$.

2. Hence $a_n \to 0$, so for all sufficiently large $|n|$ we have $|a_n| < 1/2$.

a convergent series has terms tending to $0$.

3. For $|a_n| < 1/2$, $|\log(1 + a_n)| \le 2|a_n|$.

$|\log(1+z)| \le 2|z|$ for $|z| \le 1/2$ (standard estimate).

4. Hence $\sum_{n} |\log(1 + a_n)| \le 2\sum_n |a_n| < \infty$.

3 and 1.

5. Therefore $\sum_n \log(1 + a_n)$ converges absolutely, so the product $\prod_n (1 + a_n)$ converges to a finite nonzero value.

a product $\prod (1 + a_n)$ converges (absolutely) iff $\sum \log(1 + a_n)$ converges (absolutely).

6. Q.E.D.

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