midterm1-sol - ∞ X 1 2 · 5 · 8 · ··· · (3 n + 2) 2...

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MATH 102 MIDTERM I-Solutions Date: June 18, 2004 1. a) Evaluate Z 1 dx x ( x + 1) . Putting u = x gives Z 1 dx x ( x + 1) = 2 Z 1 du 1 + u 2 = 2 (arctan u | 1 ) = π 2 . b) Test the following integral for convergence Z 0 1 + sin 2 x e x + x 2 + 1 dx. Let f ( x ) = 1 + sin 2 x e x + x 2 + 1 and g ( x ) = e - x . Then f ( x ) g ( x ) = 1 + sin 2 x 1 + x 2 e - x + e - x < 2, so 0 f ( x ) < 2 g ( x ). Since Z 0 g ( x ) dx converges, the given integral also converges by direct comparison. 2. Find the sum X 4 1 n 2 - n - 2 . Let S n = n X k =4 1 n 2 - n - 2 = n X k =4 ± 1 / 3 n - 2 - 1 / 3 n + 1 . Then S n = 1 3 ± 13 12 - 1 n - 1 - 1 n - 1 n + 1 . The sum is then found as lim n →∞ S n = 13 36 . 3. Determine if each of the following series is convergent or divergent. a) X 3 ln ( lnn ) ln 2 n Since lim n →∞ lnln n = , we have ln ln n > 1 for large n . Similarly since lim n →∞ (ln n ) 2 n = 0, we have (ln n ) 2 < n for large n . Then ln ( lnn ) ln 2 n > 1 n , and the series diverges by comparing with the harmonic series.
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b) X 1 ( n 3 + 1 - n 3 ) . ( n 3 + 1 - n 3 ) = ( n 3 + 1 - n 3 ) n 3 + 1 + n 3 n 3 + 1 + n 3 = 1 n 3 + 1 + n 3 < 1 2 n 3 = 1 2 n 3 / 2 . Then the series converges by direct comparison with the converging p-series. 4. Find the radius of convergence of the series
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Unformatted text preview: ∞ X 1 2 · 5 · 8 · ··· · (3 n + 2) 2 · 4 · 6 · ··· · (2 n ) (2 x ) n . a n = 2 · 5 · 8 ··· (3 n + 2) 2 · 4 ··· (2 n ) (2 x ) n . lim n →∞ fl fl fl fl a n +1 a n fl fl fl fl = lim n →∞ fl fl fl fl (3 n + 5)2 x 2 n + 2 fl fl fl fl = 3 | x | . For convergence we need 3 | x | &lt; 1, so the radius of convergence is 1 / 3. 5. Evaluate the limit lim x → sin ( x 2 )-x 2 cosx ( e x 2-1) x 2 . Using the Taylor expansions of sin t , cos t and e t we get lim x → sin ( x 2 )-x 2 cosx ( e x 2-1) x 2 = lim x → ( x 2-x 6 / 6 + ··· )-( x 2-x 4 / 2 + ··· ) x 4 + x 8 / 2 + ··· = lim x → 1 / 2 + x ( junk ) 1 + x ( another junk ) = 1 2 ....
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This note was uploaded on 09/28/2010 for the course MATH 1002 taught by Professor Sertäoz during the Spring '04 term at Bilkent University.

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midterm1-sol - ∞ X 1 2 · 5 · 8 · ··· · (3 n + 2) 2...

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