Uniform convergence of $\prod\limits_{n=1}^{\infty} \left (1 - \frac {z} {n} \right )$ on compact sets.

Question

Does the infinite product $\prod\limits_{n=1}^{\infty} \left (1 - \frac {z} {n} \right )$ converge uniformly on compact sets?

I know that a necessary and sufficient condition for the convergence of an infinite product which is that the required product converges if and only if $\sum\limits_{n=1}^{\infty} \log \left (1 - \frac {z} {n} \right )$ converges. But I don't have any idea about the sum of the logarithm. Another simple sufficient condition is that $\sum\limits_{n=1}^{\infty} \frac {|z|} {n}$ converges uniformly on compact sets which doesn't hold in this case since the harmonic series $\sum\limits_{n=1}^{\infty} \frac {1} {n}$ diverges. So how to deal with this problem? Any cooperation would be highly solicited.

Thanks in advance.

Rabin.

Answer 1

This may answer you with proper substitution