Find the approximation of $\sqrt{80}$ with an error $\lt 0.001$

Use $\sqrt{80}=9\sqrt{1-\frac{1}{81}}$. Now the Taylor series converges much faster: you only need $$\sqrt{1-x}=1-\frac12x+O(x^2)$$

We get $$9\left(1-\frac12\cdot\frac{1}{81}\right)=9-\frac{1}{18}=8.94444\ldots$$ with an error of $0.00017\ldots$

Tags:

Approximation

Real Analysis

Taylor Expansion

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