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Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.8
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Obtain the sum of 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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The given series is a telescoping series. (See Table 8.2.2.) The th partial sum is then
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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Context Panel: Conversions≻To List
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Context Panel: Approximate≻5 (digits)
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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 8.2.8(a) shows the convergence of the first 15 members of the sequence of partial sums to .
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>
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use plots in
module()
local Sk,X,Y,p1,p2,p3,k;
unassign('S');
Sk:=k->sum(sin(1/n)-sin(1/(n+1)),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(sin(1),k=0..15,color=black);
p3:=display(p1,p2);
print(p3)
end module:
end use:
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Figure 8.2.8(a) Convergence of to
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