Determine the radius and interval of convergence for each of the following power series. ∑n=0∞2n(x−3)nn+3

Question

Determine the radius and interval of convergence for each of the following power series.  ∑n=0∞2n(x−3)nn+3

AGRFTr · Accepted Answer

Given:The power series is:∑n=0∞an=∑n=0∞2n(x−3)nn+3For the given power series, we must ascertain the radius and interval of convergence.We know that,Whenlimn→∞|an+1an|&amp;lt;1Once the power series converges, we must determine the interval for x such that it does.We have,an=2n(x−3)nn+3&amp;nbsp;and&amp;nbsp;an+1=2(n+1)(x−3)(n+1)(n+1)+3Now,limn→∞|an+1an|&amp;lt;1limn→∞|2(n+1)(x−3)(n+1)(n+1)+32n(x−3)nn+3|&amp;lt;1limn→∞|2(x−3)n+3n+4|&amp;lt;1limn→∞|2(x−3)1+3n1+4n|&amp;lt;1|2(x−3)|&amp;lt;1−1&amp;lt;2(x−3)&amp;lt;1−12−12+352The interval for the radius for convergence is&amp;nbsp;(52,72)Now consider,For&amp;nbsp;x=3&amp;nbsp;the radius of convergence is:∑n=0∞an=∑n=0∞2n(x−3)nn+3We get, R is the radius of the convergence which is equal to zeroHence, the radius of the convergence is zero for&amp;nbsp;x=3&amp;nbsp;and the interval for the convergence is(52,72)

Jeffrey Jordon · Accepted Answer

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Determine the radius and interval of convergence for each of the following power series. sum_{n=0}^inftyfrac{2^n(x-3)^n}{sqrt{n+3}}

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