Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
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Example 8.3.1
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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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Since as , it would seem that comparing this series to the convergent -series might be a useful ploy, in which case consider
By part (1) of the Limit-Comparison test (Table 8.3.1), both series either converge or diverge. But the -series converges, so the given series converges. Since the given series has only positive terms, the convergence is absolute.
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Maple Solution
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As per the Mathematical Solution above, the absolute convergence of the given series can be established by the Limit-Comparison test, by a comparison to the convergent -series . Absolute convergence can also be established by the Integral test, but the integration is not elementary.
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Figure 8.3.1(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
Context Panel; Approximate≻5 (digits)
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>
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module()
local F,p;
F:=arctan(x)/(x^2+4);
p:=plot([F,diff(F,x)],x=1..15,color=[red,green],view=[0..15,default],tickmarks=[15,default],labels=[x,y]);
print(p);
end module:
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Figure 8.3.1(a) Graph of (red) and (green)
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The "0.I" appended to is an artifact of the numeric approximation of the exact answer Maple finds for this integral. In essence, the value of the integral is a real, positive number, and by the terms of the Integral test, the given series converges absolutely.
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