Section 2.1 of Burden and Faires: Bisection

From www.norsemathology.org

Convergence of Bisection

Theorem 2.1 Suppose that $f \in C[a,b]$ and $\left.f(a)*f(b)<0\right.$ . The bisection method generates a sequence $\{r_n\}_{n=1}^\infty$ approximating a root $\left.r\right.$ of $\left.f\right.$ with

$|r_n-r| \le \frac{b-a}{2^n}$ ,

when $n\ge 1$ .

So, using the terminology of section 1.3, we'd say that the sequence $\{r_n\}_{n=1}^\infty$ converges to $\left.r\right.$ as $O\left(\frac{1}{2^n}\right)$ with constant $\left.\kappa=b-a\right.$ .