test2review-solutions

test2review-solutions - suleimenov (bs26835) –...

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Unformatted text preview: suleimenov (bs26835) – 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. Thursday’s 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 you’re trying to say. 001 0.0 points Determine the interval of convergence of the series ∞ summationdisplay k =0 k 4 3 k (4 x − 5) k . 1. interval convergence = parenleftBig 1 2 , 2 parenrightBig correct 2. interval convergence = parenleftBig − 1 2 , 1 2 parenrightBig 3. series converges only at x = 5 4 4. interval convergence = ( −∞ , ∞ ) 5. interval convergence = parenleftBig − 2 , − 1 2 parenrightBig Explanation: The given series has the form ∞ summationdisplay k = 0 c k ( x − a ) k with c k = k 4 parenleftBig 4 3 parenrightBig k , a = 5 4 . But then, lim k →∞ c k +1 c k = lim k →∞ 4 3 parenleftBig k + 1 k parenrightBig 4 = 4 3 . By the Ratio Test, the series thus (i) converges when | x − a | < 3 4 , (ii) diverges when | x − a | > 3 4 . Now at the point x − a = 3 4 the series reduces to ∞ summationdisplay k = 0 k 4 , while at x − a = − 3 4 it reduces to ∞ summationdisplay k = 0 ( − 1) k k 4 . But in both cases these series diverge by the Divergent Test. Consequently, the interval of convergence of the given series is ( a − 3 4 , a + 3 4 ) = parenleftBig 1 2 , 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 = 7 , c 2 n +1 = 2 ( n ≥ 0) . (i) Find the interval of the convergence of the series. 1. interval of convergence = ( − 1 , 1) cor- rect 2. interval of convergence = ( − 2 , 2) 3. interval of convergence = [ − 7 , 7) 4. interval of convergence = [ − 1 , 1) 5. interval of convergence = [ − 2 , 2) 6. interval of convergence = ( − 7 , 7) Explanation: From the definition of the coefficients c n we see that f ( x ) = 7 + 2 x + 7 x 2 + 2 x 3 + . . . . suleimenov (bs26835) – 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 ( ∗ ) 7 + 7 x 2 + 7 x 4 + . . . = ∞ summationdisplay n = 0 7 x 2 n and ( ‡ ) 2 x + 2 x 3 + 2 x 5 + . . . = ∞ summationdisplay n = 0 2 x 2 n +1 separately. But ∞ summationdisplay n = 0 7 x 2 n = ∞ summationdisplay n = 0 ar n is an infinite geometric series with a = 7 , r = x 2 ....
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This note was uploaded on 09/15/2011 for the course M 55565 taught by Professor Rusin during the Spring '11 term at University of Texas.

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test2review-solutions - suleimenov (bs26835) –...

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