Find the Taylor series centered at zero for the function f(x)=ln⁡(2+x2). Determine the radius of convergence of this series.

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SkladanH · Accepted Answer

Find the Taylor series centered at zero for the function f(x)=ln⁡(2+x2). Determine the radius of convergence of this series. Consider a given function is f(x)=ln⁡(2+x2). Now, f(x)=ln⁡[2(1+x22)] =ln⁡(2)+ln⁡(1+x22) =ln⁡(2)+∑n=1∞(−1)n+1(x22)nn...[ln⁡(1+x)=∑n=1∞(−1)n+1xnn] =ln⁡(2)+∑n=1∞(−1)n+1x2nn2n Hence, the required Taylor&#039;s series of f(x)=ln⁡(2+x2) is ln⁡(2)+∑n=1∞(−1)n+1x2nn2n Consider a Taylor&#039;s series is ln⁡(2)+∑n=1∞(−1)n+1x2nn2n To find a Taylor&#039;s series, consider a sequence &amp;lt;(−1)n+1x2nn2n&amp;gt; Now, by ratio test, L=limn→∞|an+1an| =limn→∞|(−1)n+1x2n+2(n+1)2n+1(−1)n+1x2nn2n| =limn→∞|n2nx2n+2(n+1)2n+1x2n| =limn→∞|nx2(n+1)2| =|x22|  The series is convergent if L&amp;lt;1. Hence, |x22|&amp;lt;1  It implies that 0  Check at the end point x=2. Then,  ∑n=1∞(−1)n+1x2nn2n=∑n=1∞(−1)n+1(2)2nn2n  =∑n=1∞(−1)n+1n  It is convergent series.  Hence, the radius of convergence is 0

Jeffrey Jordon · Accepted Answer

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Find the Taylor series centered at zero for the function f(x)=ln(2 + x^2). Determine the radius of convergence of this series.

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