Exam 1-solutions - Version 034 – Exam 1 – mann...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Version 034 – Exam 1 – mann – (54675) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the n th term, a n , of an infinite series ∑ ∞ n =1 a n when the n th partial sum, S n , of the series is given by S n = 2 n n + 1 . 1. a n = 5 2 n 2. a n = 5 2 n 2 3. a n = 1 n 2 4. a n = 2 n ( n + 1) correct 5. a n = 1 n 6. a n = 5 n ( n + 1) Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a 1 = S 1 , a n = S n- S n- 1 ( n > 1) . But S n = 2 n n + 1 = 2- 2 n + 1 . Thus a 1 = 1, while a n = 2 n- 2 n + 1 , ( n > 1) . Consequently, a n = 2 n- 2 n + 1 = 2 n ( n + 1) for all n . 002 10.0 points Determine whether the series ∞ summationdisplay n =0 1 √ n + 4 cos nπ is conditionally convergent, absolutely con- vergent or divergent. 1. divergent 2. absolutely convergent 3. conditionally convergent correct Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as the alternating series ∞ summationdisplay n =0 (- 1) n 1 √ n + 4 = ∞ summationdisplay n = 0 (- 1) n f ( n ) with f ( x ) = 1 √ x + 4 . Now f ( n ) = 1 √ n + 4 > 1 √ n + 5 = f ( n + 1) for all n , while lim n →∞ f ( n ) = lim n →∞ 1 √ n + 4 = 0 . Consequently, by the Alternating Series Test, the given series converges. On the other hand, by the Limit Comparison Test and the p-series test with p = 1 / 2, we see that the series ∞ summationdisplay n =0 f ( n ) is divergent. Consequently, the given series is conditionally convergent . 003 10.0 points Version 034 – Exam 1 – mann – (54675) 2 Which one of the following properties does the series ∞ summationdisplay n =1 (- 1) n- 1 n 2 + 5 3 n have?...
View Full Document

This note was uploaded on 11/17/2010 for the course PHY 56630 taught by Professor Coker during the Spring '10 term at University of Texas.

Page1 / 7

Exam 1-solutions - Version 034 – Exam 1 – mann...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online