Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
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Example 8.4.19
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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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The endpoints of the interval of convergence are then , or and .
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At the right endpoint , , so the given power series becomes , which converges absolutely by the Limit-Comparison test, if a comparison is made with the convergent p-series ().
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At the left endpoint , , so the given power series becomes the alternating series , which also converges absolutely by the Limit-Comparison test, if a comparison is made with the convergent p-series ().
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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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Limit-Comparison with
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Calculus palette: Limit template
Context Panel: Evaluate and Display Inline
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The endpoints of the interval of convergence are then , or and .
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At the right endpoint , , so the given power series becomes , which converges absolutely by the Limit-Comparison test, if a comparison is made with the convergent p-series ().
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At the left endpoint , , so the given power series becomes the alternating series , which also converges absolutely by the Limit-Comparison test, if a comparison is made with the convergent p-series ().
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Hence, the interval of convergence is .
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Assuming , Maple sums this series to
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Figure 8.4.19(a) is a graph of this function on the interval of convergence.
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Figure 8.4.19(a) Graph of the sum of the series
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