Homework 14

# Homework 14 - Rehman (aar638) – HW14 – sachse –...

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: Rehman (aar638) – HW14 – sachse – (56620) 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 10.0 points Which, if any, of the following statements are true? A. If summationdisplay a n is divergent, then summationdisplay | a n | is divergent. B. The Ratio Test can be used to determine whether summationdisplay 1 /n ! converges. C. If lim n →∞ a n = 0, then summationdisplay a n converges. 1. none of them 2. all of them 3. A and C only 4. B and C only 5. A only 6. A and B only correct 7. C only 8. B only Explanation: A. True: if summationdisplay | a n | were convergent, then summationdisplay a n would be absolutely convergent, hence convergent. B. True: when a n = 1 /n !, then vextendsingle vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle vextendsingle = 1 n + 1-→ as n → , ∞ , so summationdisplay a n is convergent by Ratio Test. C. False: when a n = 1 /n , then lim n →∞ a n = 0, but ∞ summationdisplay n = 1 a n = ∞ summationdisplay n =1 1 n diverges by the Integral Test. 002 10.0 points Which one of the following properties does the series ∞ summationdisplay k =3 (- 1) k − 1 k- 1 k 2 + k- 2 have? 1. absolutely convergent 2. divergent 3. conditionally convergent correct Explanation: The given series has the form ∞ summationdisplay k = 3 (- 1) k − 1 k- 1 k 2 + k- 2 = ∞ summationdisplay k = 3 (- 1) k − 1 f ( k ) where f is defined by f ( x ) = x- 1 x 2 + x- 2 . Notice that x 2 + x- 2 > 0 on [3 , ∞ ), so the terms in the given series are defined for all k ≥ 3. On the other hand, x- 1 > 0 on (1 , ∞ ), so x > 1 = ⇒ f ( x ) > . Now, by the Quotient Rule, f ′ ( x ) = ( x 2 + x- 2)- ( x- 1)(2 x + 1) ( x 2 + x- 2) 2 =- x 2- 2 x + 1 ( x 2 + x- 2) 2 ; Rehman (aar638) – HW14 – sachse – (56620) 2 in particular, f is decreasing on [3 , ∞ ). Thus by the Limit Comparison Test and the p-series Test with p = 1, we see that the series ∞ summationdisplay k =3 f ( k ) diverges, so the given series fails to be abso- lutely convergent. But k ≥ 3 = ⇒ f ( k ) > f ( k + 1) , while lim x →∞ f ( x ) = 0 . Consequently, by The Alternating Series Test, the given series is conditionally convergent . 003 10.0 points Determine which, if any, of the series A. ∞ summationdisplay m = 3 m + 2 ( m ln m ) 2 B. 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . are convergent. 1. A only 2. neither of them 3. B only 4. both of them correct Explanation: A. Convergent: use Limit Comparison Test and Integral Test with f ( x ) = 1 x (ln x ) 2 . B. Convergent: given series is a geometric se- ries ∞ summationdisplay n =0 ar n with a = 1 and r = 1 2 < 1....
View Full Document

## This note was uploaded on 02/01/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas.

### Page1 / 13

Homework 14 - Rehman (aar638) – HW14 – sachse –...

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

View Full Document
Ask a homework question - tutors are online