If $\sum a_n, \sum a_n^2, \dots , \sum a_n^{k-1}$ and $\sum |a_n|^k$ are all convergent than $\prod (1+a_n)$ is also convergent.

Question

I want to show the following:

If $\sum a_n, \sum a_n^2, \dots , \sum a_n^{k-1}$ and $\sum |a_n|^k$ are all convergent than $\prod (1+a_n)$ is also convergent.

I found this statement: $\prod (1+a_n)$ converges iff $ \sum a_n$ converges

My problem is from the Appendix of "Special Functions" of Andrews.

I am confused by the problem since I think that the statement under the link is not true. Shouldnt it be that $\sum a_n$ has to be absolutely convergent?

Back to my problem: I have no idea why he has to climd to $k$ and thant take the absolute value in order to get convergence of the product. Any help?

Answer 1

Hint

First, by ignoring eventually the first few terms you can assume WLOG that $a_n \in (-1,1)$.

$$\ln (\prod_{n=1}^N (1+a_n))=\sum_{n=1}^N \ln(1+a_n) $$

Now, approximate $\ln(1+x)$ by its $k-1$ Taylor polynomial, and show that the error in the approximation is bounded by $x^k$.