Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.12
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Use Maple to sum the series and show that the sum is the limit of the sequence of partial sums.
Note that although (partial fractions), this is not a telescoping series.
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Solution
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Preliminary analysis
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Control-drag the summand of the series.
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Context Panel: Conversions≻Partial Fractions≻
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Verify that this is not a telescoping series
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Write the denominators of the two partial fractions
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Context Panel: Sequence≻
Set to in the dialog that opens
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None of the integers in the denominators of the two partial fractions match, so there will be no pairwise cancellation. This is not a telescoping series.
Maple is essential in the following calculations because no technique for summing this series has yet been developed in this course. Moreover, the general partial sum is given in terms of , the digamma (or Psi) function that is the derivative of the log of the gamma function , itself a generalization of the factorial.
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 8.2.12(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;
unassign('S');
Sk:=k->sum(1/(9*n^2-1),n=1..k);
X:=[seq(k,k=1..15)];
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..0.2]);
p2:=plot(1/2-(1/18)*Pi*sqrt(3),k=0..10,color=black);
p3:=display(p1,p2);
print(p3)
end module:
end use:
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Figure 8.2.12(a) Convergence of to
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