Taken from Lang's R & F Analysis (p.60). For some reason I can't see why, for $t \in [0,1]$, the following is true for all natural numbers $n$ (by an inductive argument):$$0 \leq \sqrt{t} - P_n(t) \leq \frac{2\sqrt{t}}{2 + n\sqrt{t}},$$where$$P_{n+1}(t) = P_n(t) +\frac{1}{2}\left(t-P_n(t)^2\right),$$and $P_0(t) = 0$.
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