Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Example 8.2.15
Obtain the Cauchy product of the absolutely convergent series ∑n=1∞1n2 and the conditionally convergent series ∑n=1∞−1n+1n.
Is the product the product of the sums of the two 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
Define bn as a piecewise function of n
bn={0n=0−1n+1nn≥1→assign as functionb
Generate the sequence cnn=050
Use the sum and seq commands.
C≔seqsumak⋅bn−k,k=0..n,n=0..50:
Obtain the partial sums Sk=∑n=0kcn,k=0,…,50
Use the add and seq commands.
S__k≔seqaddCk,k=1..n,n=1..51:
From Example 8.2.2, ∑n=1∞1n2=π2/6, and from Example 8.2.16, ∑n=1∞−1n+1n=ln2, so the Cauchy product should have the value π2 ln2/6≐1.1402. Figure 8.2.15(a) displays the partial sums just computed, along with the line y=π2 ln2/6. This figure gives evidence tha the Cauchy product does indeed sum to the "right" value.
use plots in module() local A,C,B,X,SK,p1,p2,p3,n,k; A:=n->piecewise(n=0,0,n>=1,1/n^2); B:=n->piecewise(n=0,0,n>=1,(-1)^(n+1)/n); C:=[seq(sum(A(k)*B(n-k),k=0..n),n=0..50)]; SK:=[seq(add(C[k],k=1..n),n=1..51)]: X:=[seq(k,k=0..50)]; p1:=pointplot(X,SK,symbol=solidcircle,symbolsize=15,color=red); p2:=plot(Pi^2/6*ln(2),k=0..50); p3:=display(p1,p2); print(p3); end module: end use:
Figure 8.2.15(a) Convergence of Sk to S=π2 ln2/6
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