ReviewTest3_F07Solution

ReviewTest3_F07Solution - Review Problems Test 3-Solutions...

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Review Problems – Test 3-Solutions 1. Find the interval of convergence of each of the following power series: a) 1 ( 2) ( 1) 2 n n n n x n = b) 0 (3 2) 1 n n x n = + 1 1 0 0 ( 2) 2 ( 2) ( 2) 2 2 2 lim lim lim ( 1)2 ( 2) ( 1)2 2 ( 2) 2 1 2 2 2 So the series converges if 1 1 1 2 2 2 0 4 2 2 Check endpoints: 4 : 2 1 ( 1) ( 1) which 2 n n n n n n n n n n n n n n n n n x n x x n x n x n x n x n x x x x x n n + + →∞ →∞ →∞ = = = = = + + + < ⇒ − < < ⇒ − < < < < = = 0 0 is the convergent alternating harmonic series. 0 : ( 2) 1 ( 1) which diverges. 2 Interval of convergence: (0,4] n n n n n x n n = = = = b) 1 1 (3 2) 1 (3 2) (3 2) 1 lim lim 2 (3 2) 2 (3 2) (3 2) 1 lim 3 2 1 1 3 2 1 2 1 1 1 3 3 1 so the radius of convergence is 1/3. 3 Check endpoints: 1 1 : ( 1) which converges 3 1 n n n n n n n n n x n x x n n x n x x n x x n x x x n + →∞ →∞ →∞ = + + = = + + + = < ⇒ − < < + < < < < = + 1 by the Alternating Series Test. 1 1: which diverges. 1 Interval of convergence: 1 [ ,1) 3 n x n = = + 2. Find the sum of the geometric series or show sum does not exist: a)
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