By expanding the integrand in a power series, calculate
By expanding the integrand in a power series, calculate $\displaystyle\int_0^1 e^{-x^2}\,dx$ with accuracy $10^{-3}$.
Explanation of Method
Substituting $t=-x^2$ into the exponential series and integrating term by term gives an alternating series; the Leibniz bound tells us exactly how many terms are needed for the requested tolerance.
- Substitute into the $e^t$ series and integrate term by term.
- Sum terms until the next term is below $10^{-3}$.
- Confirm the error bound and compare to the true value.
Step 1 · Expand and integrate
$$ e^{-x^2} = \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{n!} \implies \int_0^1e^{-x^2}\,dx = \sum_{n=0}^{\infty}\frac{(-1)^n}{n!(2n+1)}. $$
Step 2 · Sum terms
$$ \begin{aligned} n=0:&\ 1.000000 & n=3:&\ -0.023810 \\ n=1:&\ -0.333333 & n=4:&\ 0.004630 \\ n=2:&\ 0.100000 & n=5:&\ -0.000758 \end{aligned} $$ Sum through $n=5$: $1-0.333333+0.1-0.023810+0.004630-0.000758=0.746729$.
Step 3 · Error bound and comparison
The next term, $n=6$: $\dfrac1{6!\cdot13}=\dfrac1{9360}\approx0.000107<10^{-3}$, so the partial sum is accurate to within the required tolerance. Rounded to the requested precision: $$ \int_0^1e^{-x^2}\,dx \approx 0.747. $$ (The true value is $0.746824\ldots$, well within the $10^{-3}$ tolerance.)
Summary Table
| Property | Value |
|---|---|
| Series | $\sum(-1)^n/[n!(2n+1)]$ |
| Result | $0.747$ |
| Related function | Error function, $\operatorname{erf}(1)$ |
Additional Notes
- This integral cannot be expressed in elementary functions; it is proportional to the error function, $\int_0^1e^{-x^2}dx=\frac{\sqrt\pi}2\operatorname{erf}(1)$, central to the normal distribution in probability theory.
- The alternating-series bound (as in Problem 279) directly certifies the stopping point: six terms guarantee $10^{-3}$ accuracy without needing to compute the true value independently.
- Compare with Problem 259's Maclaurin series for $e^{-x^3}$: the same substitution technique applies to any polynomial exponent, only the resulting integrated-series denominators change.