- Establish the following properties of infinite products.
Suppose \(a_n\neq -1\) for every \(n\). Show that if \(\sum |a_n|^2\) converges, then the product \(\prod (1+a_n)\) converges to a non-zero limit if and only if \(\sum a_n\) converges.
Find an example of a sequence of complex numbers \(\{a_n\}\) such that \(\sum a_n\) converges but \(\prod(1+a_n)\) diverges.
Also find an example such that \(\prod(1+a_n)\) converges to a non-zero limit and \(\sum a_n\) diverges.
Solution
For part (a), because \(\sum |a_n|^2<\infty\), we have \(a_n\to0\). Hence for all sufficiently large \(n\), say \(|a_n|\le1/2\), we may use the principal logarithm and the expansion \[ \log(1+a_n)=a_n+O(|a_n|^2), \tag{1} \] with an absolute implied constant. Since \(a_n\ne-1\), no factor vanishes.
The tail product \(\prod(1+a_n)\) converges to a nonzero limit if and only if the series \[ \sum \log(1+a_n) \] converges. By (1), the difference between this logarithmic series and \(\sum a_n\) is absolutely convergent because \(\sum|a_n|^2<\infty\). Therefore \[ \prod(1+a_n)\text{ converges to a nonzero limit} \iff \sum a_n\text{ converges}. \]
For part (b), take \[ a_n=\frac{(-1)^n}{\sqrt n}. \] The series \(\sum a_n\) converges by the alternating-series test. On the other hand, \[ \log(1+a_n)=a_n-\frac{a_n^2}{2}+O(|a_n|^3). \] The series \(\sum a_n\) converges and \(\sum |a_n|^3\) converges, but \[ \sum a_n^2=\sum\frac1n=\infty. \] Hence \[ \sum_{n=1}^N\log(1+a_n)\longrightarrow-\infty, \] so the partial products tend to \(0\). Thus the infinite product does not converge to a nonzero limit.
For part (c), let \[ b_n=\frac{(-1)^n}{\sqrt n}, \qquad a_n=e^{b_n}-1. \] Then \[ 1+a_n=e^{b_n}, \] so \[ \prod_{n=1}^N(1+a_n) =\exp\!\left(\sum_{n=1}^N b_n\right) \] converges to a nonzero limit because \(\sum b_n\) converges. But \[ a_n=b_n+\frac{b_n^2}{2}+O(|b_n|^3). \] Again \(\sum b_n\) and \(\sum|b_n|^3\) converge, while \[ \frac12\sum b_n^2=\frac12\sum\frac1n \] diverges to \(+\infty\). Therefore \(\sum a_n\) diverges.