Which type of singularity does this complex function have at z = ∞?

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I'm trying to classify the type of singularity at z = ∞ (the point at infinity) of the complex function:

click to see the equation

Up to now, I've just been able to prove that ∞ is not a pole. So I would like to prove that it is either an essential singularity or a removable singularity. Thank you for your suggestions!

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At , $z=n$ for each $1\neq n\in\mathbb{N}$ , $f(z)$ have zeros.

As, limit point of zeros is a isolated essential singularity.

So, $z=\infty$ is a isolated essential singularity of $f(z)$.