Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Example 8.2.12
Use Maple to sum the series ∑n=1∞19 n2−1 and show that the sum is the limit of the sequence of partial sums.
Note that although 19 n2−1=12 13 n−1−13 n+1 (partial fractions), this is not a telescoping series.
Solution
Preliminary analysis
Control-drag the summand of the series.
Context Panel: Conversions≻Partial Fractions≻n
19 n2−1= convert to partial fractions in n −123n+1+123n−1
Verify that this is not a telescoping series
Write the denominators of the two partial fractions
Context Panel: Sequence≻n Set n=1 to n=10 in the dialog that opens
3 n−1→sequence w.r.t. n2,5,8,11,14,17,20,23,26,29
3 n+1→sequence w.r.t. n4,7,10,13,16,19,22,25,28,31
None of the integers in the denominators of the two partial fractions match, so there will be no pairwise cancellation. This is not a telescoping series.
Maple is essential in the following calculations because no technique for summing this series has yet been developed in this course. Moreover, the general partial sum is given in terms of ψx, the digamma (or Psi) function that is the derivative of the log of the gamma function Γx, itself a generalization of the factorial.
Obtain the sum of the series
Control-drag the series.
Context Panel: Evaluate and Display Inline
∑n=1∞19 n2−1 = 12−118π3
Obtain an expression for the kth partial sum
Control-drag the series and change ∞ to k.
Context Panel: Assign to a Name≻S[k]
∑n=1k19 n2−1 = 16Ψk+23−16Ψk+43−118π3+12→assign to a nameSk
Display the first few partial sums
Type Sk and press the Enter key.
Context Panel: Sequence≻k In the resulting dialog box, set k=1 to k=10
Sk
16Ψk+23−16Ψk+43−118π3+12
→sequence w.r.t. k
18,43280,93560,1385980080,28433160160,934401951731680,946159151731680,10984293015949143200,11066012015949143200,10007836228995348279736800
Obtain the limit of the partial sums
Calculus palette: Limit template≻Apply to Sk
limk→∞Sk = 12−118π3
Figure 8.2.12(a) shows the convergence of the first 10 members of the sequence of partial sums to S=12−118π3≐0.19769.
use plots in module() local Sk,X,Y,p1,p2,p3,k; unassign('S'); Sk:=k->sum(1/(9*n^2-1),n=1..k); X:=[seq(k,k=1..15)]; Y:=[seq(Sk(k),k=1..10)]; p1:=pointplot(X,Y,symbol=solidcircle,symbolsize=15,color=red,labels=[k,typeset(S[k])],view=[0..10,0..0.2]); p2:=plot(1/2-(1/18)*Pi*sqrt(3),k=0..10,color=black); p3:=display(p1,p2); print(p3) end module: end use:
Figure 8.2.12(a) Convergence of Sk to S≐0.19769
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