Prove that infinite complex product diverges

Question

How can one prove that $\prod_{n=1}^{\infty} (1 + \frac{i}{n})$ diverges? I tried with considering $log(P_n)$, where $P_n$ is the $n$-th partial product. Then $log(P_n)= \sum_{k=1}^{n} (1+\frac{i}{k}) $. Can we proceed by finding Taylor series for logarithmic function and how? Thanks a lot in advance.

Answer 1

Hint: $\text{Log}\left(1+\dfrac{i}{n}\right)=\ln\left|1+\dfrac{i}{n}\right|+i\text{Arg}\left(1+\dfrac{i}{n}\right)=\ln\left|1+\dfrac{i}{n}\right|+\underbrace{\arctan\left(\frac{1}{n}\right)}_{\sim1/n}$.
Observe that $\prod_{j=1}^{+\infty}|1+i/n|$ is convergent.