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could someone please help me with this exercise?

Let $$f_n=\frac{\log\left(z-\frac{1}{n}\right)}{\log^\frac{3}{2}(n)\left(z^2+n\right)}$$ Where $\log(z)$ is the natural branch of the function logarithm.
Find the holomorphy domain of $f(z)=\sum\limits_{n=2}^{\infty} f_n$.
I tried to use the fact that a series of holomorphic functions witch converge nomally to a function $f$ implies the holomorphy of $f$ but i can't find a majoration.
I also would be happy to recive other approaches to this kind of exercises.

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Hint: For each $n$ determine where $f_n$ is well defined. This will be a set of the form $\mathbb C\setminus E_n.$ The set where $\sum f_n$ makes sense can be no larger than $\mathbb C\setminus (\cup \,E_n).$

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  • $\begingroup$ Thank you, $E_n$ should be {$z\in \mathbb{C}:z\neq i\sqrt{n},$ $z\neq-i\sqrt{n}$ } $\cup$ $ ${$z\in \mathbb{C}: Re(z)\leq \frac{1}{n}$ $and $ $ Img(z)=0$}. $\endgroup$
    – yo yo
    Jan 7 '19 at 22:10
  • $\begingroup$ I would write $E_n = \{i\sqrt n,-i\sqrt n\}\cup (-\infty,1/n].$ $\endgroup$
    – zhw.
    Jan 7 '19 at 23:26

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