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

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

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.
