Infinite product convergence

Question

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges.

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I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ converges.

Answer 1

Note: $|1+\frac{i}{n}| = (1+\frac{1}{n^2})^{1/2}$

$\prod_n |1+\frac{i}{n}|$ converges $\Leftrightarrow \sum_n \log{|1+\frac{i}{n}|} = \frac{1}{2}\sum_n \log{(1+\frac{1}{n^2})}$ converges

which is true if $ \sum|\frac{1}{n^2}|$ converges.

Answer 2

We can see the divergence of the first series by

$$\log\left(1+\frac{i}{n}\right)\sim_\infty\frac i n$$

moreover we have $$\log\left\vert 1+\frac{i}{n}\right\vert=\frac{1}{2}\log\left(1+\frac 1 {n^2}\right)\sim_\infty\frac{1}{2n^2}$$ so the second series is convergent.