Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
Example 8.4.22
Determine the radius of convergence and the interval of convergence for the power series ∑k=0∞−1k x2 k+nk! k+n! 22 k+n.
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.
Solution
Mathematical Solution
Since the given power series contains the powers x2 k, the radius of convergence R is given by
R2=limk→∞ak/ak+1 = limk→∞k+1! k+1+n! 22k+1+nk! k+n! 22 k+n=limk→∞4k+1k+1+n=∞
Hence, the interval of convergence is −R,R=−∞,∞.
The given power series defines the special function BesselJn,x, often written with the notation Jnx. It is defined as the solution of Bessel's differential equation of order n, and plays an important role in the solution of many applied problems in physics and engineering.
Maple Solution
Define the general coefficient an as a function of n
Write an=… Context Panel: Assign Function
ak=1k! k+n! 22 k+n→assign as functiona
Obtain the radius of convergence
Calculus palette: Limit template Context Panel: Assign Name
R=limk→∞akak+1→assign
Display R, the radius of convergence
Write R Context Panel: Evaluate and Display Inline
R = ∞
That Maple can sum this power series to Jnx is shown by the following calculation. Notice how the gamma function Γn appears, and is related to the factorial function for n an integer.
Control-drag the given power series. Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Assuming Integer
∑k=0∞−1k x2 k+nk! k+n! 22 k+n = BesselJn,xΓn+1n!→assuming integerBesselJn,x
Figure 8.4.22(a) contains graphs of Jnx for n=0,1,2, colored respectively, black, red, green.
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])]);
Figure 8.4.22(a) Graphs of J0x (black), J1x (red), J2x (green)
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