Question

Does arctan(5n) converge or diverge?

Answer 1

If you mean, "does the sequence $a_n=\arctan(5n)$ converge", then yes, since $\lim\limits_{n\to\infty}\arctan n=\dfrac\pi2$.

If you mean, "does the series $\sum\limits_n\arctan(5n)$ converge", then no, for the reason above (the summand passes the nth limit test for divergence, so the series diverges).