HW12-solutions

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: adams (bda255) – HW12 – Radin – (56635) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 3) 10.0 points If the n th partial sum of ∑ ∞ n = 1 a n is S n = 2 n − 3 n + 1 , (i) what is a 1 ? 1. a 1 = − 1 2 correct 2. a 1 = 1 2 3. a 1 = 2 4. a 1 = 5 2 5. a 1 = − 5 2 Explanation: Since a 1 = S 1 , a 1 = − 1 2 . 002 (part 2 of 3) 10.0 points (ii) What is a n for n > 1? 1. a n = 5 n ( n + 1) correct 2. a n = 5 n 2 3. a n = 1 n 2 4. a n = 1 n ( n + 1) 5. a n = 5 n ( n − 1) 6. a n = 1 n ( n − 1) Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a n = S n − S n − 1 . But S n = 2 n − 3 n + 1 = 2( n + 1) − 5 n + 1 = 2 − 5 n + 1 . Consequently, a n = 5 n − 5 n + 1 = 5 n ( n + 1) for all n > 1. 003 (part 3 of 3) 10.0 points (iii) What is the sum ∑ ∞ n = 1 a n ? 1. sum = 5 2. sum = 2 correct 3. sum = 3 4. sum = 4 5. sum = 6 Explanation: By definition sum = lim n →∞ S n = lim n →∞ parenleftBig 2 n − 3 n + 1 parenrightBig . Thus sum = 2 . 004 10.0 points Determine whether the series ∞ summationdisplay n =1 7 − n 8 n +1 is convergent or divergent. If it is convergent, find its sum. 1. 82 adams (bda255) – HW12 – Radin – (56635) 2 2. 56 3. 37 4. 40 5. divergent correct Explanation: ∞ summationdisplay n =1 7 − n 8 n +1 = ∞ summationdisplay n =1 8 parenleftbigg 8 7 parenrightbigg n . Since 8 7 > 1, the series is divergent. 005 10.0 points Find the sum of the series ∞ summationdisplay n = 0 (4 x − 3) n 2 n for those values of x for which it converges. 1. sum = − 2 1 − 4 x 2. sum = 5 − 4 x 2 3. sum = − 2 1 + 4 x 4. sum = 2 5 + 4 x 5. sum = − 1 + 4 x 2 6. sum = 2 5 − 4 x correct Explanation: When the geometric series ∑ ∞ n =0 ar n con- verges it has sum = a 1 − r . Now for the given series, a = 1 , r = 4 x − 3 2 , Consequently, it has sum = 1 1 − 4 x − 3 2 = 2 5 − 4 x . 006 10.0 points Find the sum of the infinite series tan 2 θ − tan 4 θ + tan 6 θ + ... + ( − 1) n − 1 tan 2 n θ + ... whenever the series converges. 1. sum = sin 2 θ correct 2. sum = − cos 2 θ 3. sum = cos 2 θ 4. sum = − sin 2 θ 5. sum = tan 2 θ Explanation: For general θ the series tan 2 θ − tan 4 θ + tan 6 θ + ... + ( − 1) n − 1 tan 2 n θ + ... is an infinite geometric series whose common ratio is − tan 2 θ . Since the initial term in this series is tan 2 θ , its sum is thus given by tan 2 θ 1 + tan 2 θ = tan 2 θ sec 2 θ . Consequently sum = sin 2 θ . 007 10.0 points A business executive realizes that he is out of shape so he begins an exercise program in adams (bda255) – HW12 – Radin – (56635) 3 which he jogs daily and each day thereafter jogs 7% more miles than he did on the previ- ous day. The program will be complete when he has jogged a total of at least 66 miles. If he jogs 5 miles on the first day, what is the mini- mum number of days he will have to exercise...
View Full Document