Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Expand the integrand in ∫01sinx2 ⅆx and integrate termwise to obtain an estimate of the integral guaranteed correct to three decimal places.
Compare to the value Maple provides.
Maple can evaluate the given integral exactly in terms of the special function FresnelS. This exact solution can then be approximated to a sufficiently high accuracy.
∫01sinx2 ⅆx = 12⁢FresnelS⁡2π⁢2⁢π≐0.3102683013
Fresnel is the name of a French physicist. The correct pronounciation of his name can be heard at a number of internet sites, including the one at the end of this link.
Expanding the integrand in a Maclaurin series gives
The results of termwise integration of the series expansion are listed in Table 8.5.11(a). Because the resulting series is alternating, the remark in Table 8.2.2 applies, that is, the error in a partial sum is less than the first neglected term. Table 8.5.11(a) suggests that just the first three terms need to be added for the approximation to be accurate to three decimal places.
∫01.x2 ⅆx = 0.3333333333
∫01.x6/3! ⅆx = 0.02380952381
∫01.x10/5! ⅆx = 0.0007575757576
∫01.x14/7! ⅆx = 0.00001322751323
Table 8.5.11(a) Termwise integration
Obtain an accurate value for the given integral
Control-drag the integral.
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻10 (digits)
∫01sinx2 ⅆx = 12⁢FresnelS⁡2π⁢2⁢π→at 10 digits0.3102683013
Expand the integrand in a Maclaurin series
Write the integrand.
Context Panel: Series≻Series≻x
See Figure 8.5.11(a).
Figure 8.5.11(a) Series dialog
Implement the calculations in Table 8.5.11(a), reproduced here for convenience. Note that the upper limit of integration is written as a floating-point number, which then causes Maple to evaluate the integral numerically.
sinx2→series in xx2−16⁢x6+1120⁢x10−15040⁢x14
Sum the first three terms of the integrated series
∫01.x2 ⅆx−∫01.x6/3! ⅆx+∫01.x10/5! ⅆx = 0.3102813853
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