First we observe that `1/(1-text[(]-t^2text[)]) = 1/(1 + t^2)`, so the substitution in `1/(1 - x)` produces the right function. The same substitution in the geometric polynomial of degree `n` gives `1 - t^2 + t^4 - cdots +text[(]-1text[)]^nt^(2n)`.
For values of `t` between `-1` and `1`, `-t^2` is also in this interval, so we would expect the polynomials to approximate `1text[/(]1 + t^2text[)]` if `text[|] t text[|] < 1`. On the other hand, if `text[|] t text[|]>= 1`, then `|-t^2| >= 1` also, so the polynomials cannot be expected to approximate outside the interval `text[|] t text[|]< 1`.
Note that the substitution turns a polynomial of degree `n` (in the variable `x`) into a polynomial of degree `2n` in the variable `t`. Indeed, only even-degree terms appear in these polynomials, which is appropriate, because `1text[/(]1 + t^2text[)]` is an even function. When we integrate term-by-term, we get polynomials with only odd-degree terms — which is appropriate, because arctan`text[(] x text[)]` is an odd function:
`int_0^x [1 - t^2 + t^4 - cdots + text[(]-1text[)]^n t^(2n)] dt = x - 1/3x^3 + 1/5x^5 - cdots + (text[(]-1text[)]^n)/(2n + 1) x^(2n + 1)`.
These polynomials should approximate `text[arctan(] x text[)].`