Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Expand the integrand in ∫01J0x ⅆx and integrate termwise to obtain an estimate of the integral guaranteed correct to three decimal places.
Compare to the value Maple provides.
The notation J0x represents the special function Maple knows as BesselJ. The full syntax for this function would be BesselJ0,x, but by invoking special typesetting rules through the Typesetting Rules Assistant, (View menu, Typesetting Rules) Maple can access this Bessel function with the simpler syntax J0x.
Maple evaluates the given integral numerically, giving ∫01.0J0x ⅆx≐0.9197304101. Expanding the integrand in a Maclaurin series gives
The results of termwise integration of the series expansion are listed in Table 8.5.12(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 8.5.12(a) suggests that just the first three terms need to be added for the approximation to be accurate to three decimal places.
∫01.01 ⅆx = 1.0
∫01.0x2/4 ⅆx = 0.08333333333
∫01.0x4/64 ⅆx = 0.003125000000
∫01.0x6/2304 ⅆx = 0.00006200396825
Table 8.5.12(a) Termwise integration
Obtain an accurate value for the given integral
Write the integral in terms of BesselJ0,x.
Context Panel: Evaluate and Display Inline
∫01.0BesselJ0,x ⅆx = 0.9197304101
Expand the integrand in a Maclaurin series
Write the integrand in terms of BesselJ0,x.
Context Panel: Series≻Series≻x
See Figure 8.5.12(a).
Figure 8.5.12(a) Series dialogc
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.
BesselJ0,x→series in x1−14⁢x2+164⁢x4−12304⁢x6+1147456⁢x8
Sum the first three terms of the integrated series
∫01.01 ⅆx−∫01.0x2/4 ⅆx+∫01.0x4/64 ⅆx = 0.9197916667
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