Find a power series for the function, centered at c, and determine the interval of convergence.&nbsp;g(x)=32x-1,c=2

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Find a power series for the function, centered at c, and determine the interval of convergence.&amp;nbsp;g(x)=32x-1,c=2

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&amp;nbsp;Geometric Power Series Centered at c: The geometric power series centered at c is a series of the forma1−(r−c)=∑n=0∞a(r−c)n,|r−c|&amp;lt;1where a is the first term and r-c is the common ratio.The function is given byf(x)=32x−1,c=2Writing f(x) in the form&amp;nbsp;a1−r&amp;nbsp;produces32x−1=32(x−−2)+3=33(1+23(x−2))=1(1+23(x−2))=a1−rwhich implies that a=1 and&amp;nbsp;r=−23(x−2).So, the power series for f(x) is32x−1=∑n=0∞arn=∑n=0∞[−23(x−2)]n=∑n=0∞(−23)n(x−2)n=1−23(x−2)+49(x−2)2−827(x−2)3+…When the power series converges,23|x−2|&amp;lt;1⇒|x−2|&amp;lt;32⇒−32&amp;lt;x−2&amp;lt;32⇒12&amp;lt;x&amp;lt;312This suggests that the convergence interval is&amp;nbsp;(12,312)Whenever the power series converges23|x−2|&amp;lt;1⇒|x−2|&amp;lt;32⇒−32&amp;lt;x−2&amp;lt;322−32&amp;lt;x&amp;lt;2+32&amp;nbsp;

Find a power series for the function, centered at c, and determine the interval

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