Binomial Series — Question 7

*Question 7 Approximate $\sqrt{1.04}$ using only the constant and linear terms of the binomial series for $\sqrt{1+x}$. Use the alternating-series structure to bound the omitted correction and report the approximation.

*Question 7 – Solution

$$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\cdots.$$ Set $x=0.04$.

$$\sqrt{1.04}\approx1+\frac{0.04}{2}=\boxed{1.02}.$$

Beginning with the quadratic term, the omitted terms alternate and decrease in magnitude at $x=0.04$. Therefore $$|R|\le\frac{(0.04)^2}{8}=\boxed{0.0002}.$$ The first correction is negative, so the linear approximation is an overestimate. More precisely, $$1.0198\le\sqrt{1.04}<1.02.$$ The actual value, $1.0198039\ldots$, is consistent with this bound.