HW 12 key

# HW 12 key - Miller Kierste – Homework 12 – Due 3:00 am...

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

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: Miller, Kierste – Homework 12 – Due: Apr 17 2007, 3:00 am – Inst: Gary Berg 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points If a n , b n , and c n satisfy the inequalities < b n ≤ c n ≤ a n , for all n , what can we say about the series ( A ) : ∞ X n = 1 a n , ( B ) : ∞ X n = 1 b n if we know that the series ( C ) : ∞ X n = 1 c n is convergent but know nothing else about a n and b n ? 1. ( A ) converges , ( B ) converges 2. ( A ) converges , ( B ) need not converge 3. ( A ) converges , ( B ) diverges 4. ( A ) diverges , ( B ) converges 5. ( A ) diverges , ( B ) diverges 6. ( A ) need not converge , ( B ) converges correct Explanation: Let’s try applying the Comparison Test: (i) if < b n ≤ c n , X n c n converges , then the Comparison Test applies and says that X b n converges; (ii) while if < c n ≤ a n , X n c n converges , then the Comparison Test is inconclusive be- cause X a n could converge, but it could di- verge - we can’t say precisely without further restrictions on a n . Consequently, what we can say is ( A ) need not converge , ( B ) converges . keywords: Comparison Test, conceptual 002 (part 1 of 1) 10 points Determine whether the series ∞ X n = 1 tan- 1 n 2 + n 5 converges or diverges. 1. series is divergent 2. series is convergent correct Explanation: We apply the Limit Comparison Test with a n = tan- 1 n 2 + n 5 , b n = 1 n 5 . For lim n →∞ n 5 ‡ tan- 1 n 2 + n 5 · = lim n →∞ tan- 1 n = π 2 . Thus the given series ∞ X n = 1 tan- 1 n 2 + n 5 is convergent if and only if the series ∞ X n = 1 1 n 5 Miller, Kierste – Homework 12 – Due: Apr 17 2007, 3:00 am – Inst: Gary Berg 2 is convergent. But by the p-series test, this last series converges because p = 5 > 1. Con- sequently, the given series is convergent . keywords: 003 (part 1 of 1) 10 points Determine whether the series ∞ X n = 1 2 + n + n 2 √ 4 + n 2 + n 6 converges or diverges. 1. series is convergent 2. series is divergent correct Explanation: We apply the Limit Comparison Test with a n = 2 + n + n 2 √ 4 + n 2 + n 6 , b n = 1 n . For then lim n →∞ a n b n = lim n →∞ 2 n + n 2 + n 3 √ 4 + n 2 + n 6 = lim n →∞ 2 n 2 + 1 n + 1 r 4 n 6 + 1 n 4 + 1 = 1 > . Thus the given series ∞ X n = 1 2 + n + n 2 √ 4 + n 2 + n 6 converges if and only if the series ∞ X n = 1 1 n converges. But, by the p-series test (or be- cause the harmonic series diverges), this last series diverges because p = 1. Consequently, the series is divergent . keywords: 004 (part 1 of 1) 10 points Which of the following series converge(s)?...
View Full Document

## This note was uploaded on 05/08/2008 for the course M 408 L taught by Professor Cepparo during the Spring '08 term at University of Texas.

### Page1 / 14

HW 12 key - Miller Kierste – Homework 12 – Due 3:00 am...

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

View Full Document
Ask a homework question - tutors are online