Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
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Example 8.3.6
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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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Mathematical Solution
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The series is alternating, suggesting the Leibniz test. For this, must eventually become a monotone decreasing sequence with limit zero. Consequently, the following calculations are executed.
= and for , or
The conditions of the Leibniz test apply, and by that test, the series converges conditionally.
To test for absolute convergence, apply the Integral test: = .
By this test, the series of absolute values does not converge, so the series is not absolutely convergent.
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Maple Solution
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Figure 8.3.6(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 as a test for absolute convergence, provided the integration starts from, say, .
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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,N;
F:=ln(x)/sqrt(x);
N:=15;
p:=plot([F,diff(F,x)],x=1..N,color=[red,green],view=[0..N,default],tickmarks=[N,default],labels=[x,y]);
print(p);
end module:
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Figure 8.3.6(a) Graph of (red) and (green)
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Since the integral diverges, the series does not converge absolutely.
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To establish the conditional convergence by the Leibniz test, implement the following two calculations.
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Calculus palette: Limit operator
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Context Panel: Evaluate and Display Inline
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=
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Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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=
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A modicum of algebra puts the derivative into the form , from which it can be seen that the derivative is negative for , or . (This is consistent with Figure 8.3.6(a).) Since the criteria of the test are satisfied, it follows that the series converges conditionally.
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