Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.14
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Obtain the Cauchy product of with itself. Is the value of the product the square of the value of the 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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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, , so that the Cauchy product should have the value ≐ . Figure 8.2.14(a) displays the partial sums just computed, along with the line . This figure gives evidence that the Cauchy product does indeed sum to the "right" value.
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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:
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Figure 8.2.14(a) Convergence of to
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