Is there any sequence satisfies this :$\exp\left(\sum_{n\geq0}a_n\right)=\sum_{n\geq0}\exp(a_n)$ for which both becames convergents?

Question

I seek an example of a sequence $a_n $ numerical or integers for which both side of (1) is convergent. Is there any sequence satisfies this:$$ \exp\left(\displaystyle \sum_{n\geq0}a_n\right)=\sum_{n\geq0}\exp(a_n)\tag{1}$$

What is an example of a sequence which satisfies $(1)$? If no such sequence exists, how can one prove that no such sequence exists?