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=\frac\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).