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).
