Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
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Example 8.4.14
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Determine the radius of convergence and the interval of convergence for the power series .
Even though (7) in Table 8.4.1 claims that absolute convergence at one end of the interval of convergence implies absolute convergence at the other, if the convergence at an endpoint is absolute, verify that it also absolute at the other.
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Solution
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Mathematical Solution
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Since the given power series contains the powers , the radius of convergence is given by
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=
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At the right endpoint , the given power series becomes , which diverges by the Integral test since decreases monotonically to zero on and its integral on that interval diverges.
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At the left endpoint , the given power series becomes the alternating series , which converges conditionally by the Leibniz test since the sequence decreases monotonically to zero.
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Hence, the interval of convergence is .
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Maple Solution
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In Figure 8.4.14(a), the graph of
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is drawn in red, while the graph of its derivative
is drawn in green. Since for , it sould be clear that decreases monotonically to zero as .
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>
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module()
local f,p;
f:=1/x/ln(x);
p:=plot([f,diff(f,x)],x=2..10,color=[red,green],labels=[x,y],tickmarks=[8,3]);
print(p);
end module:
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Figure 8.4.14(a) Graph of (red) and (green)
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Define the general coefficient as a function of
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Write
Context Panel: Assign Function
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Obtain the radius of convergence
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Calculus palette: Limit template
Context Panel: Assign Name
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Display , the radius of convergence
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Write
Context Panel: Evaluate and Display Inline
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Test for convergence at
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Calculus palette: Definite integral template
Context Panel: Evaluate and Display Inline
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Test for convergence at
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Write
Context Panel: Evaluate and Display Inline
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=
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At the right endpoint , the given power series becomes , which diverges by the Integral test. Figure 8.4.14(a) and the related discussion reveal that , and is an appropriate function to use as an integrand. Since the integral diverges, so too does the series.
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At the left endpoint , the given power series becomes the alternating series , which converges conditionally by the Leibniz test since the sequence decreases monotonically to zero.
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Hence, the interval of convergence is .
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