Meromorphic function with given principle part Need to find meromorphic function with poles at $z=\sqrt{n}$ and principle part is $\frac{1}{(z-\sqrt{n})^4}+\frac{1}{z-\sqrt{n}}$ . It's clear that the first term would converge uniformly on the compact subsets of $\mathbb{C}$ because in comparison with $\frac{1}{n^2}$. But adding $\frac{1}{\sqrt{n}}$ is not changing the convergence of the second term. Any hints regarding how to claim the convergence of the series ?

The general idea is to develop the wanted principal parts into a Taylor series and take “sufficiently many” terms to that the rest converges locally uniformly.
In our case we have
$$
 \frac{1}{z-\sqrt{n}} = -\frac{1}{\sqrt{n}} \left( 1 + \frac{z}{\sqrt n} + \frac{z^2}{n} + \cdots\right)
$$
so that
$$
\frac{1}{z-\sqrt{n}} + \frac{1}{\sqrt n} + \frac{z}{n} = O\left( \frac{z^2}{n^{3/2}}\right)
$$
This shows that
$$
 \sum_{n=1}^\infty \frac{1}{z-\sqrt{n}} + \frac{1}{\sqrt n} + \frac{z}{n}
$$
converges uniformly on each compact set not containing any of the poles. Combining that with your result we get
$$
 f(z) = \sum_{n=1}^\infty \frac{1}{(z-\sqrt{n})^4} + \frac{1}{z-\sqrt{n}} + \frac{1}{\sqrt n} + \frac{z}{n}
$$
as a meromorphic function with the desired poles and principal parts.
