Represent the function&nbsp;3ln⁡(8−x)&nbsp;as a power series&nbsp;What is the radius of convergence?

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Represent the function&amp;nbsp;3ln⁡(8−x)&amp;nbsp;as a power series&amp;nbsp;What is the radius of convergence?

yunitsiL · Accepted Answer

To write the function f(x)=3ln⁡(8−x) as a power series (Maclaurin series). Also to calculate the radius of convergence. From the standard power series, the Maclaurin series of ln⁡(1+x) is given by, ln⁡(1+x)=∑n=1∞(−1)n−1xnn, |x|&amp;lt;1 Here, |x|&amp;lt;1 is its radius of converhence Rewrite the function as shown below f(x)=3ln⁡(8−x) f(x)=3ln⁡(8(1−x8)) f(x)=3ln⁡(8)+3ln⁡(1−x8) Now, ln⁡(1−x8)=∑n=1∞(−1)n−1(−x8)nn,|−x8|&amp;lt;1 ln⁡(1−x8)=∑n=1∞(−1)n−1(−1)nxnn8n|x|8&amp;lt;1 ln⁡(1−x8)=∑n=1∞−(−1)2nxnn8n|x|&amp;lt;8 ln⁡(1−x8)=∑n=1∞−xnn8n|x|&amp;lt;8 Use it above f(x)=3ln⁡(8)−3∑n=1∞xnn8n This is the required power series and the radius of convergence is 8 (from |x|&amp;lt;8) Answer: Power series: f(x)=3ln⁡(8)−3∑n=1∞xnn8n The radius of convergence is 8.

Jeffrey Jordon · Accepted Answer

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Represent the function 3ln(8-x) as a power series (Maclaurin series) What is the radius of convergence?

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