Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
Example 8.4.21
Determine the radius of convergence and the interval of convergence for the power series ∑n=2∞n!2 xn2 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 xn, the radius of convergence is given by
R=limn→∞an/an+1 = limn→∞n!2/2 n!n+1!2/2n+1!=limn→∞22n+1n+1=4
At the right endpoint x=4, the given power series becomes Σ an 4n, which diverges by the nth-term test because limn→∞an4n=∞.
At the left endpoint x=−4, the given power series becomes the alternating series Σ −1nan 4n, which also diverges by the nth-term test because limn→∞an4n=∞.
Hence, the interval of convergence is −R,R=−4,4.
Maple Solution
Define the general coefficient an as a function of n
Write an=… Context Panel: Assign Function
an=n!22 n!→assign as functiona
Obtain the radius of convergence
Calculus palette: Limit template Context Panel: Assign Name
R=limn→∞anan+1→assign
Display R, the radius of convergence
Write R Context Panel: Evaluate and Display Inline
R = 4
Test for convergence at x=R=4
Calculus palette: Limit template Context Panel: Evaluate and Display Inline
limn→∞an⋅4n = ∞
Maple can actually sum this series - Figure 8.4.21(a) is a graph of this function on the interval of convergence.
Enter the series. Context Panel: Simplify≻Assuming Real Range≻−4<x<4
module() local S,p; S:=sum((n!)^2/(2*n)!*x^n,n=2..infinity) assuming abs(x)<4; p:=plot(S,x=-4..4,0..20,labels=[x,y]); print(p); end module:
Figure 8.4.21(a) Graph of the sum of the series
∑n=2∞n!2 xn2 n!→assuming real range12x−x5/2+8arcsin12x4−x+6x3/2−8x−4+x2
<< Previous Example Section 8.4 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
