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Unformatted text preview: hussain (tsh476) test2review rusin (55565) 1 This print-out should have 38 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Again this review is optional and non- graded. Thursdays exam will focus on mate- rial since the last test, hence these problems are limited to those topics. Please remember that the exam format is not like the Quest for- mat: give yourself some practice writing out your solutions in a way that makes it clear what youre trying to say. 001 0.0 points Determine the interval of convergence of the series summationdisplay k =0 k 3 2 k (3 x 4) k . 1. interval convergence = parenleftBig 2 3 , 2 3 parenrightBig 2. interval convergence = parenleftBig 2 , 2 3 parenrightBig 3. series converges only at x = 4 3 4. interval convergence = ( , ) 5. interval convergence = parenleftBig 2 3 , 2 parenrightBig correct Explanation: The given series has the form summationdisplay k = 0 c k ( x a ) k with c k = k 3 parenleftBig 3 2 parenrightBig k , a = 4 3 . But then, lim k c k +1 c k = lim k 3 2 parenleftBig k + 1 k parenrightBig 3 = 3 2 . By the Ratio Test, the series thus (i) converges when | x a | < 2 3 , (ii) diverges when | x a | > 2 3 . Now at the point x a = 2 3 the series reduces to summationdisplay k = 0 k 3 , while at x a = 2 3 it reduces to summationdisplay k = 0 ( 1) k k 3 . But in both cases these series diverge by the Divergent Test. Consequently, the interval of convergence of the given series is ( a 2 3 , a + 2 3 ) = parenleftBig 2 3 , 2 parenrightBig 002 (part 1 of 2) 0.0 points A function f is defined by the series f ( x ) = summationdisplay n = 0 c n x n in which the coefficients c n are specified by c 2 n = 5 , c 2 n +1 = 3 ( n 0) . (i) Find the interval of the convergence of the series. 1. interval of convergence = [ 1 , 1) 2. interval of convergence = [ 3 , 3) 3. interval of convergence = ( 1 , 1) cor- rect 4. interval of convergence = [ 5 , 5) 5. interval of convergence = ( 3 , 3) 6. interval of convergence = ( 5 , 5) Explanation: From the definition of the coefficients c n we see that f ( x ) = 5 + 3 x + 5 x 2 + 3 x 3 + . . . . hussain (tsh476) test2review rusin (55565) 2 Now the sum summationdisplay n = 0 ( a n + b n ) of two convergent series is again convergent, so consider the series ( ) 5 + 5 x 2 + 5 x 4 + . . . = summationdisplay n = 0 5 x 2 n and ( ) 3 x + 3 x 3 + 3 x 5 + . . . = summationdisplay n = 0 3 x 2 n +1 separately. But summationdisplay n = 0 5 x 2 n = summationdisplay n = 0 ar n is an infinite geometric series with a = 5 , r = x 2 ....
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This note was uploaded on 04/03/2011 for the course MATH 408D taught by Professor Chu during the Spring '09 term at University of Texas at Austin.

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REVIEW2 - hussain (tsh476) test2review rusin (55565) 1 This...

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