Applications of Series — Question 10

*Question 10 Construct the quadratic Maclaurin model for the Gaussian $e^{-x^2}$. Determine where that polynomial is positive, and give an error bound for $|x|\le1$ using the first omitted term.

*Question 10 – Solution

$$e^{-x^2}=1-x^2+\frac{x^4}{2!}-\frac{x^6}{3!}+\cdots.$$ Keeping terms through degree $2$ gives $$\boxed{Q_2(x)=1-x^2}.$$

$$Q_2(x)>0\iff1-x^2>0\iff |x|<1.$$ It vanishes at $x=\pm1$ and becomes negative for $|x|>1$, whereas the true Gaussian is positive for every real $x$. This shows the model is local.

For $|x|\le1$, the series in $x^2$ alternates with decreasing magnitudes. Therefore $$\boxed{|e^{-x^2}-(1-x^2)|\le\frac{x^4}{2}}.$$ The error is order $x^4$ near the origin.