Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.15
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Obtain the Cauchy product of the absolutely convergent series and the conditionally convergent series .
Is the product the product of the sums of the two given series?
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Solution
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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 as a piecewise function of
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Expression palette: Piecewise template
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Context Panel: Assign Function
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Define as a piecewise function of
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Expression palette: Piecewise template
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Context Panel: Assign Function
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Generate the sequence
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Use the sum and seq commands.
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Obtain the partial sums
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Use the add and seq commands.
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From Example 8.2.2, , and from Example 8.2.16, , so the Cauchy product should have the value . Figure 8.2.15(a) displays the partial sums just computed, along with the line . This figure gives evidence tha the Cauchy product does indeed sum to the "right" value.
>
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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:
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Figure 8.2.15(a) Convergence of to
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