Calculus II · Power series and Taylor series
Integrate a Gaussian series with a certified error
Problem
Approximate \[ G=\int_0^{0.4}e^{-x^2}\,dx \] by replacing $e^{-x^2}$ with $1-x^2+x^4/2-x^6/6$. Give both an exact expression and a decimal value for the approximation. Then use the alternating-series structure on $0\le x\le0.4$ to prove that the absolute error is at most \[ \frac{0.4^9}{216}. \] State a decimal value for this bound and identify the first omitted term after integration.
Hint
Integrate $x^{2n}$ as $x^{2n+1}/(2n+1)$ before substituting $0.4$.
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