Answer
Diverges
Work Step by Step
Given $$ \sum_{n=1}^{\infty}\lim_{n\to\infty} \sqrt{4 n^{2}+1}-n$$
Since
\begin{align*}
\lim_{n\to\infty} \sqrt{4 n^{2}+1}-n &=
\lim_{n\to\infty} \frac{\sqrt{4 n^{2}+1}-n}{\sqrt{4 n^{2}+1}+n} \times \sqrt{4 n^{2}+1}+n\\
&=\lim_{n\to\infty} \frac{3 n^{2}+1}{\sqrt{4 n^{2}+1}+n}\\
&= \lim_{n\to\infty} \frac{\left(3 n+\frac{1}{n}\right)}{\sqrt{4+\frac{1}{n^{2}}}+1} \\
&=\infty
\end{align*}
Then the series diverges.
