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**Def.**Let $\{z_j\}$ be a sequence of non-zero complex numbers. We call the exponent of convergence of the sequence the positive number $b$, if it exists,

$$b=inf\{\rho >0 :\sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\rho}}<\infty \}$$

Now consider the function

$$f(z)=e^{e^z}-1$$

Find the zeros $\{z_j\}$ of $f$ and their exponent of convergence.

This is what i did: we have to solve the equation:

$$e^{e^z}=1$$

thus

$$e^z=2n_1\pi i$$

for $n_1$ integer not zero.

We have

$$z=log(2n_1\pi i)+2n_2\pi i$$

with $n_2$ integer.

Now i think the exponent is $\infty$, as we have a $log$, but i don't know how to formalize it. In particular, i have the problem to enumerate the $z_j's$, since each of them has two integers $n_1,n_2$ and i can't see how to express

$$\sum_{j=1}^{+\infty}\frac{1}{|z_j|}$$

Any help would be appreciated!

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