I am trying to answer the following question: Suppose that $(a_n)_{n=1}^\infty$ is a sequence of nonzero complex numbers and $k$ is a positive integer such that the series $$\sum_{n=1}^\infty \left( a_n - \frac{1}{2}a_n^2 + \cdots + \frac{(-1)^k-1}{k}a_n^k \right)$$ and $$\sum_{n=1}^\infty |a_n|^k$$ are convergent in $\mathbb{C}$ and convergent in $\mathbb{R}$ respectively. Prove or disprove that $\prod_{n=1}^\infty (1+a_n)$ converges in $\mathbb{C}$. What I have worked out:
Since $\sum_{n=1}^\infty |a_n|^k$ is convergent, so the sequence $$\left( \sum_{n=1}^\infty |a_n|^j \right)_{j=k+1}^\infty$$ is a monotone decreasing sequence of reals that converges to $0$ as $j$ tends to infinity. Hence by the Alternating Series test, $$\sum_{j=k+1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^\infty |a_n|^j$$ is a convergent series. Now, let a positive integer $N$ be given. Then $$\sum_{n=1}^N \left( \log(1+a_n)- \left( a_n - \frac{1}{2}a_n^2 + \cdots + \frac{(-1)^k-1}{k}a_n^k \right) \right) $$ $$= \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^N a_n^j$$ and the latter series converges in $\mathbb{C}$ when $N$ tends to infinity because $\sum_{j=k+1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^\infty |a_n|^j$ is convergent in $\mathbb{R}$. Since both the series $$\sum_{j=1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^\infty a_n^j$$ and $$\sum_{n=1}^\infty \left( a_n - \frac{1}{2}a_n^2 + \cdots + \frac{(-1)^k-1}{k}a_n^k \right)$$ are convergent in $\mathbb{C}$, we conclude that so is $\sum_{n=1}^\infty \log(1+a_n)$, and hence so is $\prod_{n=1}^\infty (1+a_n)$.
Would appreciate any comments on the following: Is it valid to leap from $\sum_{j=k+1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^\infty |a_n|^j$ convergent in $\mathbb{R}$ to $\sum_{j=k+1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^\infty a_n^j$ convergent in $\mathbb{C}$? And if this conceptual leap is invalid, how should I rectify this proof? And is this $$\sum_{n=1}^N \left( \log(1+a_n)- \left( a_n - \frac{1}{2}a_n^2 + \cdots + \frac{(-1)^k-1}{k}a_n^k \right) \right) = \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j} \sum_{n=1}^N a_n^j$$ valid as well?
