Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Example 8.2.2
Use Maple to sum the convergent p-series ∑n=1∞1/n2 and show that the sum is the limit of the sequence of partial sums.
Solution
Obtain the sum of the series
Control-drag the given series.
Context Panel: Evaluate and Display Inline
∑n=1∞1/n2 = π26
Obtain the partial sum Sk and the first few values of Sk
Expression palette: Summation template Write the sum through n=k.
Context Panel: Sequence≻k In the pop-up dialog: from k=1 to k=10
∑n=1k1/n2 = −Ψ1,k+1+π26→sequence w.r.t. k1,54,4936,205144,52693600,53693600,266681176400,1077749705600,97781416350400,19683291270080
Obtain the limit of the sequence of partial sums
Calculus palette: Limit operator
Expression palette: Summation template Sum up through n=k.
limk→∞∑n=1k1n2 = π26
Maple expresses the partial sum via the special function Ψ, which is related to derivative(s) of the log of the gamma function, itself a generalization of the factorial function. Nevertheless, Maple's value for Sk will be accepted as correct.
Figure 8.2.2(a) shows the convergence of the first 30 members of the sequence of partial sums to π2/6≐1.6450.
use plots in module() local Sk,X,Y,p1,p2,p3,k; Sk:=k->sum(1/n^2,n=1..k); X:=[seq(k,k=1..30)]; Y:=[seq(Sk(k),k=1..30)]; p1:=pointplot(X,Y,symbol=solidcircle,symbolsize=15,color=red,labels=[k,typeset(S[k])],view=[0..10,0..2]); p2:=plot(Pi^2/6,k=0..30,color=black); p3:=display(p1,p2); print(p3) end module: end use:
Figure 8.2.2(a) Convergence of Sk to S=π2/6 ≐1.6450
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