# HW 14 - Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00...

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: Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 1 This print-out should have 17 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 Compare the radius of convergence, R 1 , of the series ∞ X n = 0 c n y n with the radius of convergence, R 2 , of the series ∞ X n = 1 n c n y n- 1 when lim n →∞ fl fl fl c n +1 c n fl fl fl = 2 . 1. R 1 = 2 R 2 = 2 2. R 1 = R 2 = 2 3. R 1 = R 2 = 1 2 correct 4. 2 R 1 = R 2 = 2 5. R 1 = 2 R 2 = 1 2 6. 2 R 1 = R 2 = 1 2 Explanation: When lim n →∞ fl fl fl c n +1 c n fl fl fl = 2 , the Ratio Test ensures that the series ∞ X n = 0 c n y n is (i) convergent when | y | < 1 2 , and (ii) divergent when | y | > 1 2 . On the other hand, since lim n →∞ fl fl fl ( n + 1) c n +1 nc n fl fl fl = lim n →∞ fl fl fl c n +1 c n fl fl fl , the Ratio Test ensures also that the series ∞ X n = 1 n c n y n- 1 is (i) convergent when | y | < 1 2 , and (ii) divergent when | y | > 1 2 . Consequently, R 1 = R 2 = 1 2 . keywords: 002 (part 1 of 1) 10 points Find a power series representation for the function f ( x ) = 1 x- 2 . 1. f ( x ) =- ∞ X n = 0 2 n x n 2. f ( x ) = ∞ X n = 0 (- 1) n- 1 2 n +1 x n 3. f ( x ) = ∞ X n = 0 (- 1) n 2 n x n 4. f ( x ) = ∞ X n = 0 1 2 n +1 x n 5. f ( x ) =- ∞ X n = 0 1 2 n +1 x n correct Explanation: We know that 1 1- x = 1 + x + x 2 + . . . = ∞ X n = 0 x n . Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 2 On the other hand, 1 x- 2 =- 1 2 ‡ 1 1- ( x/ 2) · . Thus f ( x ) =- 1 2 ∞ X n = 0 ‡ x 2 · n =- 1 2 ∞ X n = 0 1 2 n x n . Consequently, f ( x ) =- ∞ X n = 0 1 2 n +1 x n with | x | < 2. keywords: 003 (part 1 of 1) 10 points Find a power series representation for the function f ( x ) = ln(3- x ) . 1. f ( x ) = ln3 + ∞ X n = 0 x n n 3 n 2. f ( x ) = ln3- ∞ X n = 0 x n 3 n 3. f ( x ) =- ∞ X n = 1 x n n 3 n 4. f ( x ) = ln3- ∞ X n = 1 x n n 3 n correct 5. f ( x ) = ∞ X n = 0 x n n 3 n 6. f ( x ) = ln3 + ∞ X n = 1 x n 3 n Explanation: We can either use the known power series representation ln(1- x ) =- ∞ X n = 1 x n n , or the fact that ln(1- x ) =- Z x 1 1- s ds =- Z x n ∞ X n = 0 s n o ds =- ∞ X n = 0 Z x s n ds =- ∞ X n = 1 x n n . For then by properties of logs, f ( x ) = ln3 ‡ 1- 1 3 x · = ln3 + ln ‡ 1- 1 3 x · , so that f ( x ) = ln3- ∞ X n = 1 x n n 3 n . keywords: 004 (part 1 of 1) 10 points Find a power series representation centered at the origin for the function f ( y ) = y 3 (2- y ) 2 . 1. f ( y ) = ∞ X n = 3 n- 2 2 n- 1 y n correct 2. f ( y ) = ∞ X n = 2 1 2 n- 1 y n 3. f ( y ) = ∞ X n = 3 n 2 n y n 4. f ( y ) = ∞ X n = 3 1 2 n- 3 y n 5. f ( y ) = ∞ X n = 2 n- 1 2 n y n Explanation: Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00 am – Inst: Diane Radin 3 By the known result for geometric series, 1 2- y = 1 2 ‡ 1- y 2 · = 1 2 ∞ X n = 0...
View Full Document

## This note was uploaded on 04/12/2010 for the course PHY 58195 taught by Professor Turner during the Spring '09 term at University of Texas.

### Page1 / 10

HW 14 - Kim, Jin – Homework 14 – Due: Dec 4 2007, 3:00...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online