Exponential of Series Equals Infinite Product

Theorem
Let $\left\langle{z_n}\right\rangle$ be a sequence of complex numbers.

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

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

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

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

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

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

By Exponential of Complex Number is Nonzero, $\exp z \ne 0$.

Thus $\displaystyle \lim_{n \mathop \to \infty} P_n = \exp z \ne 0$.

Also see

 * Logarithm of Infinite Product of Complex Numbers