Let zn∈C∖{0}z_n \in \mathbb{C}\setminus \{0\} for all nn. Prove that ∏∞n=1zn\prod_{n=1}^{\infty}z_n converges iff ∑∞n=1Log(zn)\sum_{n=1}^{\infty}Log(z_n) converges.

One part is easy as exponential function is continuous but the other part is not able to solve ( as we know that Log is not continuous).

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how you define the complex logarithm ?
– Tsemo Aristide
Oct 20 at 18:00

  

 

By convergence, did you mean convergence to a nonzero value?
– zhw.
Oct 20 at 18:01

  

 

The domain for a branch of the logarithm is obtained by removing from the plane a ray (emanating from the origin). Since some of your znz_{n}’s may be on that ray, you need to use two branches of the logarithm, and treat the convergence issues on both of them.
– user8960
Oct 20 at 18:09

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