To prove that a coherent state is an eigenstate of the annihilation operator

[chmodfree]

The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle $$
How can I show that the state is eigenstate of annihilation operator a?
i.e.
$$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$

 

[PeterDonis]

How can I show that the state is eigenstate of annihilation operator a?

What does the annihilation operator do to each term in the sum?

 

[Demystifier]

What does the annihilation operator do to each term in the sum?

Maybe one should add that the exponential function must be expanded in the Taylor series.