Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
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Example 8.4.22
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Determine the radius of convergence and the interval of convergence for the power series .
Even though (7) in Table 8.4.1 claims that absolute convergence at one end of the interval of convergence implies absolute convergence at the other, if the convergence at an endpoint is absolute, verify that it also absolute at the other.
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Solution
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Mathematical Solution
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Since the given power series contains the powers , the radius of convergence is given by
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=
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Hence, the interval of convergence is .
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The given power series defines the special function , often written with the notation . It is defined as the solution of Bessel's differential equation of order , and plays an important role in the solution of many applied problems in physics and engineering.
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Maple Solution
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Define the general coefficient as a function of
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Write
Context Panel: Assign Function
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Obtain the radius of convergence
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Calculus palette: Limit template
Context Panel: Assign Name
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Display , the radius of convergence
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Write
Context Panel: Evaluate and Display Inline
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=
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Hence, the interval of convergence is .
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That Maple can sum this power series to is shown by the following calculation. Notice how the gamma function appears, and is related to the factorial function for an integer.
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Control-drag the given power series.
Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Integer
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=
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Figure 8.4.22(a) contains graphs of for , colored respectively, black, red, green.
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plot([BesselJ(0,x),BesselJ(1,x),BesselJ(2,x)],x=0..10,color=[black,red,green],labels=[x,y],legend=[typeset(J[0]),typeset(J[1]),typeset(J[2])]);
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Figure 8.4.22(a) Graphs of (black), (red), (green)
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