Answer
See the proof below.
Work Step by Step
We have
$$\lim _{n \to \infty}a_n=\lim _{n \to \infty} \left(b-\tan ^{-1} n^{2}\right)=b-\tan^{-1}\infty\\
=b-\frac{\pi}{2}$$
Whenever $b\neq \pi/2$, then $\lim _{n \to \infty}a_n\neq0$ and hence the series diverges (it converges only if $\lim _{n \to \infty}a_n=0$).
