Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.11
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Use Maple to sum the series and show that the sum is the limit of the sequence of partial sums.
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Solution
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Mathematical Solution
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A partial fraction decomposition gives , so that the given series is a telescoping series. (See Table 8.2.2.)
Twice the th partial sum is then
where the red terms cancel in pairs, but the two bold terms will be left in any finite sum.
Consequently, the sum of the series is given by
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Maple Solution
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Obtain the sum of the series
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Control-drag the series.
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Context Panel: Evaluate and Display Inline
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Obtain an expression for the th partial sum
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Control-drag the series and change ∞ to .
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻S[k]
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Display the first few partial sums
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Type and press the Enter key.
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Context Panel: Sequence≻
In the resulting dialog box, set to
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Obtain the limit of the partial sums
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Calculus palette: Limit template≻Apply to
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Context Panel: Evaluate and Display Inline
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Figure 8.2.11(a) shows the convergence of the first 15 members of the sequence of partial sums to .
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use plots in
module()
local Sk,X,Y,p1,p2,p3,k;
unassign('S');
Sk:=k->sum(1/n/(n+2),n=1..k);
X:=[seq(k,k=1..15)];
Y:=[seq(Sk(k),k=1..15)];
p1:=pointplot(X,Y,symbol=solidcircle,symbolsize=15,color=red,labels=[k,typeset(S[k])],view=[0..15,0..1]);
p2:=plot(3/4,k=0..15,color=black);
p3:=display(p1,p2);
print(p3)
end module:
end use:
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Figure 8.2.11(a) Convergence of to
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