Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.2
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Use Maple to sum the convergent -series and show that the sum is the limit of the sequence of partial sums.
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Solution
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Obtain the sum of the series
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Control-drag the given series.
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Context Panel: Evaluate and Display Inline
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Obtain the partial sum and the first few values of
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Expression palette: Summation template
Write the sum through .
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Context Panel: Evaluate and Display Inline
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Context Panel: Sequence≻
In the pop-up dialog: from to
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Obtain the limit of the sequence of partial sums
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Calculus palette: Limit operator
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Expression palette: Summation template
Sum up through .
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Context Panel: Evaluate and Display Inline
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Maple expresses the partial sum via the special function , 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 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 .
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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:
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Figure 8.2.2(a) Convergence of to ≐1.6450
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