calculus and analysis - Generating a polynomial that's accurate to within an error of no more than 1/10^5

I'm currently stuck on a question for class that asks...

"Find a polynomial p[x] that you can use to calculate 6 ArcTan[x] to within an error of no more than 10^(-5) for all the x's with -(1/Sqrt[3]) <= x <= 1/Sqrt[3]."

I used a series expansion below.

Clear[x];   
approx6arctan[x_] = Normal[Series[6Tan[x], {x, 0, 200}]]

However, this can only generate a function that's accurate only to the fourth decimal, no matter how much I expand the series (200 is already huge).

Any hints on how to generate a polynomial that's accurate to the fifth decimal? Thanks in advance.

Comments

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