Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.3
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a)
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Use Maple to sum the alternating series and show that the sum is the limit of the sequence of partial sums.
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b)
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Test the claim that a partial sum is closer to the sum than the magnitude of the first neglected term.
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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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Obtain the first few values of
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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
Lerch Phi. Nevertheless, Maple's value for will be accepted as correct.
Figure 8.2.3(a) shows the convergence of the first 10 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+1)/n^2,n=1..k);
X:=[seq(k,k=1..10)];
Y:=[seq(Sk(k),k=1..10)];
p1:=pointplot(X,Y,symbol=solidcircle,symbolsize=15,color=red,labels=[k,typeset(S[k])],view=[0..10,0..1]);
p2:=plot(Pi^2/12,k=0..10,color=black);
p3:=display(p1,p2);
print(p3)
end module:
end use:
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Figure 8.2.3(a) Convergence of to
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