Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Example 8.2.14
Obtain the Cauchy product of ∑n=1∞1n2 with itself. Is the value of the product the square of the value of the given series?
Solution
Because the expressions that will be generated are very large, commands have been used to generate results that could also have been generated with the Context Panel system.
Define an as a piecewise function of n
Expression palette: Piecewise template
Context Panel: Assign Function
an=0n=01n2n≥1→assign as functiona
Generate the sequence cnn=025
Use the sum and seq commands.
C≔seqsumak⋅an−k,k=0..n,n=0..25:
Obtain the partial sums Sk=∑n=0kcn,k=0,…,25
Use the add and seq commands.
S__k≔seqaddCk,k=1..n,n=1..26:
From Example 8.2.2, ∑n=1∞1n2=π2/6, so that the Cauchy product should have the value π4/36 ≐ 2.7059. Figure 8.2.14(a) displays the partial sums just computed, along with the line y=π4/36. This figure gives evidence that the Cauchy product does indeed sum to the "right" value.
use plots in module() local A,C,X,SK,p1,p2,p3,n,k; A:=n->piecewise(n=0,0,n>=1,1/n^2); C:=[seq(sum(A(k)*A(n-k),k=0..n),n=0..25)]; SK:=[seq(add(C[k],k=1..n),n=1..26)]: X:=[seq(k,k=0..25)]; p1:=pointplot(X,SK,symbol=solidcircle,symbolsize=15,color=red); p2:=plot(Pi^4/36,k=0..25); p3:=display(p1,p2); print(p3); end module: end use:
Figure 8.2.14(a) Convergence of Sk to S=π4/36
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