Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once). Give the interval of convergence for the resulting series. g(x)=ln⁡(1−2x) using f(x)=11−2x

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lobeflepnoumni · Accepted Answer

To determine the radius of convergence and derive the power series of g(x) from the power series for f(x)The geometric series generated from is all that the power series for f(x) is.&amp;nbsp;11−y, setting&amp;nbsp;y=2x. Its convergence radius is 0.5.Letf(x)=11−2x=1+(2x)+(2x)2+...+(2x)n+...expanded power series (geometric series),valid for&amp;nbsp;|2x|&amp;lt;1,&amp;nbsp;|x|&amp;lt;0.5so, radius of convergence&amp;nbsp;=0.5By term-by-term integrating the power series of f(x), one may find the power series for g(x) (upto a constant). G(dradius )&#039;s of convergence is same to f(x )&#039;s (from general theory)&amp;nbsp;=0.5Now,&amp;nbsp;g(x)=ln⁡(1−2x)=−2∫dx(1−2x)=−2∫f(x)dx=−2∫[1+(2x)+(2x)2+...+(2x)n+...]dx=−2x−(2x)22−(2x)33−(2x)44−...−(2x)nn−...is the radius of convergence for the power series expansion of g(x)&amp;nbsp;=0.5

Jeffrey Jordon · Accepted Answer

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Find the power series representation for g centered at 0 by differentiating or integrating the power series for f(perhaps more than once).

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