# Section 2.1 of Burden and Faires: Bisection

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## Convergence of Bisection

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

${\displaystyle |r_{n}-r|\leq {\frac {b-a}{2^{n}}}}$,

when ${\displaystyle n\geq 1}$.

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

In the big picture, this means that we get one additional digit of binary accuracy with each iteration of bisection.