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q5solns

# q5solns - Mathematics 104 Quiz 5 Friday April 4 2008 n 3 xn...

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Mathematics 104, Quiz 5, Friday, April 4, 2008 1. Consider the power series summationdisplay n =1 n 3 · x n 3 n . For which values of x is this series convergent? As usual, explain your reasoning. vextendsingle vextendsingle vextendsingle vextendsingle ( n + 1) 3 · x n +1 3 n +1 · 3 n n 3 · x n vextendsingle vextendsingle vextendsingle vextendsingle = parenleftbigg n + 1 n parenrightbigg 3 · | x | 3 | x | 3 as n goes to infinity since ( n + 1) /n goes to 1 by L’Hopital’s rule. So by the ratio test this series is absolutely convergent when | x | / 3 < 1 or in other words if x ( - 3 , 3). It is divergent if x > 3 or if x < - 3 by the ratio test. It remains to check the endpoints. If x = 3 then the series becomes summationdisplay 1 n 3 which diverges because it fails the nth term test since n 3 → ∞ as n goes to infinity. Similarly when x = - 3 we get the series summationdisplay 1 ( - 1) n · n 3 , which also diverges because it too fails the n th term test.

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q5solns - Mathematics 104 Quiz 5 Friday April 4 2008 n 3 xn...

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