A Sequence of Polynomials for Approximating Arctangent

k=0 (−1)k 2k + 1 x } , the sequence of Taylor polynomials centered at 0 that converges to arctanx on [−1, 1]. Like the Taylor polynomials for several other classical functions, e.g., cosx, sinx, and ex, this sequence of polynomials is very easy to describe and work with; but unlike those Taylor sequences with factorials in the denominators of their coefficients, it does not converge rapidly for all “important” values of x. In particular, it converges extremely slowly to arctanx when |x| is near 1. For example, if x = 0.95, we need to use T28, a polynomial of degree 57, to get three decimal places of accuracy for arctan(0.95); if x = 1, we need to use T500, a polynomial of degree 1001, to get three decimal places for arctan 1. Indeed, for x ∈ [0, 1], it is easy to show that | arctanx−Tn(x)| ≥ x2n+3 2(2n+3) ; thus, as x → 1, Tn(x) cannot approximate arctanx any better than 1 2(2n+3) = 1 2(degree Tn)+4 . The same is true near −1. It is only fair to note that {Tn} converges to arctanx reasonably fast for x near 0. In this note we present another elementary, easily-described sequence in Q[x] that approximates arctanx uniformly on [0, 1] and which does so much more rapidly than the sequence {Tn}. Such an approximating sequence provides, via the identities arctanx = − arctan(−x) = π2 − arctan( 1 x), a method of approximating arctanx for all x ∈ R. The approximating sequence arises from the family of rational functions {x4m(1− x)4m 1 + x2 } m∈N .