Chapter 8: Infinite Sequences and Series
Section 8.2: Series
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Example 8.2.1
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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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The geometric series sums to , and the finite sum adds to .
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Since the given series is a geometric series with , its sum is
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The partial sum up through is , which, in the limit as , becomes .
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Figure 8.2.1(a) shows the first few partial sums rapidly converging to .
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use plots in
module()
local Sk,X,Y,p1,p2,p3,k;
Sk:=k->(1-(1/3)^(k+1))/(1-1/3);
X:=[seq(k,k=0..10)];
Y:=[seq(Sk(k),k=0..10)];
p1:=pointplot(X,Y,symbol=solidcircle,symbolsize=15,color=red,labels=[k,typeset(S[k])],view=[0..10,0..2]);
p2:=plot(1.5,k=0..10,color=black);
p3:=display(p1,p2);
print(p3)
end module:
end use:
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Figure 8.2.1(a) Convergence of to
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Maple Solution
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Maple "knows" how to sum a geometric series:
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 and the first few partial sums
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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 the series to .
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Context Panel: Evaluate and Display Inline
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=
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