Exponential of Series Equals Infinite Product

Theorem
Let Let $(z_n)$ be a sequence of complex numbers.

Suppose $\displaystyle\sum_{n=1}^\infty z_n$ converges to $z\in\C$.

Then $\displaystyle\prod_{n=1}^\infty\exp(z_n)$ converges to $\exp z$.

Proof
Let $S_n$ be the $n$th partial sum of $\displaystyle\sum_{n=1}^\infty z_n$.

Let $P_n$ be the $n$th partial product of $\displaystyle\prod_{n=1}^\infty\exp(z_n)$.

By Exponential of Sum, $\exp(S_n)=P_n$ for all $n\in\N$.

By Exponential Function is Continuous, $\displaystyle\lim_{n\to\infty}\exp(S_n)=\exp z$.

By Exponential of Complex Number is Nonzero, $\exp z\neq0$.

Thus $\displaystyle\lim_{n\to\infty}P_n=\exp z\neq0$.

Also see

 * Logarithm of Infinite Product of Complex Numbers