Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$
I originally wanted to try to prove it by showing $$\lim \prod^N \exp(a_i)=\exp\left(\lim \sum^N a_i\right),$$ but it doesn't seem to work.
Thank you.
I don't think this has anything to do with the complex logarithm function. If the series $\sum a_i$ converges, then the left-hand side will exist (because $\exp$ is continuous): $\exp (\sum \limits _{i=1} ^\infty a_i) = \exp (\lim \limits _{N \to \infty} \sum \limits _{i=1} ^N a_i) = \lim \limits _{N \to \infty} \exp (\sum \limits _{i=1} ^N a_i) = \lim \limits _{N \to \infty} \prod \limits _{i=1} ^N \exp (a_i) = \prod \limits _{i=1} ^\infty \exp (a_i)$.
If the series diverges, then the left-hand side may fail to exist, while the right-hand side may have a well-defined value (understood as the limit of a sequence of finite products), exactly as you have noticed in the first version of your question (before editing it).