Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0...

Numerical analysis Taylor's method question: Find a value of $n$ necessary
for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0...

Let $f(x)=\tan^{-1}(x)$
Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$
Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within
$10^{-6}$ on $[-0.5,0.5]$.
Is it the smalled possible value?
My attempt:
$f'(x)=\dfrac{1}{x^2+1}$
$f''(x)=\dfrac{-2x}{(x^2+1)^2}$
As I attempt to find $f^n(x)$, the expression gets even more complicated.
So I try to evaluate the error term for $P_1(x)$.
Which gets me to within $0.125$.
And I am stuck