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Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
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Example 8.3.2
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Determine if the series diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
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Solution
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Divergence of this series can be established by the Integral test.
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Figure 8.3.2(a) contains a graph of the function (in red) and of its derivative (in green).
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On the basis of this graph, it may be conjectured that is monotone decreasing and bounded below by zero, provided . (The derivative appears to be negative for .)
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Consequently, the Integral test may be tried, provided the integration starts from, say, . This is a nontrivial integration, one Maple evaluates in terms of the special function .
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Calculus palette: Definite integral template
Context Panel: Evaluate and Display Inline
=
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>
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module()
local F,p;
F:=1/x/ln(x);
p:=plot([F,diff(F,x)],x=1..5,color=[red,green],view=[0..5,-10..10],tickmarks=[5,default]);
print(p);
end module:
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Figure 8.3.2(a) Graph of (red) and (green)
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Since the integral is unbounded, the series diverges.
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