Proving uniform convergence of a sequence

Question

I have to prove the uniform convergence of this sequence $f_n(x)=\tan^{-1}nx$ in $[a,b],a>0$

What I have reached so far:

$$|f_n(x)-f(x)|=\left|\tan^{-1}nx-\frac\pi 2\right|=\tan^{-1}nx-\frac\pi 2<\epsilon$$

How do I proceed further ?

Answer 1

Recalling the Taylor series of $\arctan(t)$ at $t=\infty$

$$ \frac{\pi}{2} -{\frac {1}{t}}+O \left( {t}^{-3} \right) $$

$$ \implies \arctan( nx )=\frac{\pi}{2}-{\frac {1}{ny}}+O \left( {n}^{-3} \right) $$

$$ \implies \arctan( nx )-\frac{\pi}{2}= -\frac{1}{nx}+O \left( {n}^{-3} \right) $$

$$ \implies \arctan( nx )-\frac{\pi}{2} \sim -\frac{1}{nx}. $$

Now, you can advance to finish the problem.

Answer 2

Hint: $\lim_{x\to+\infty}\tan^{-1} x=\frac\pi 2$ means that for every $\epsilon>0$ there exists $x_0$ such that $x>x_0$ implies $\left|\tan^{-1}x-\frac\pi2\right|<\epsilon$.