Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
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Example 8.4.1
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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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At the right endpoint , the given power series becomes , which converges absolutely by part (1) of the Limit-Comparison test if the comparison series is taken as the convergent p-series .
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The relevant calculation that must be made is . Since is a (finite) real number, both series converge. Hence, the power series converges absolutely at .
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At the left endpoint , the given power series becomes the alternating series , which has just been shown to converge absolutely.
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Hence, the interval of convergence is .
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Maple Solution
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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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Write
Context Panel: Assign Function
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Calculus palette: Limit template
Context Panel: Evaluate and Display Inline
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The comparison series is a convergent p-series, so by part (1) of the Limit-Comparison test, the given series converges absolutely at . The given series also converges absolutely at because = , and the series with those terms has already been shown to converge absolutely.
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Hence, the interval of convergence is .
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At the right endpoint , the resulting series yields neither to the Ratio test () nor to the Integral test (Maple cannot evaluate the resulting integral). To find an appropriate comparison series, look for the limiting behavior of , which can be seen in the dominant term of its asymptotic series, obtained via the Context Panel as follows.
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Write the expression.
Context Panel: Evaluate and Display Inline
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Context Panel: Series≻Asymptotic series≻n
(In the Series dialog that opens, set "Series order" to 2.)
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Maple's asympt command will also provide such an expansion, as shown in Table 8.4.1(a), below.
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Table 8.4.1(a) Asymptotic behavior of
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The factor in the first term of the expansion is what suggests a comparison to the p-series .
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