math128_w15_asst9sol - MATH 128 Winter 2015 Assignment 9...

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MATH 128 Winter 2015Assignment 9 Solutions1. State the centre, radius, and interval of convergence of the given power series.Xn=0((n+ 1)!)25n(x+ 1)n
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= limn→∞nn+ 1|x|=|x|3To converge we must have that|x|3<1. That is, we must have|x|<3. Thus,the radius of convergence is 3. We now check the endpointsx=±3.Atx=-3, we haveXn=1(-1)nn23n(-3)n=Xn=11n2which is a convergentp-series.Atx= 3, we haveXn=1(-1)nn23n3n=Xn=1(-1)nn2which converges by the alternatingseries test.Thus, the interval of convergence is [-3,3].(d)Xn=18nn(2x+ 1)nRe-writing the series in the formXn=1an(x-c)nwe getXn=18nn(2x+ 1)n=Xn=18nn2nx+12n=Xn=116nnx+12n
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