# Exam 4 - for all(You may use for each Solution Since Hence...

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

Next: About this document MATH 1014.03MW Exam 3 Monday, March, 6 2000 NAME: Student Number: No. Marks Instructions : This exam contains 6 questions and has a total of 100 marks. Show all of your work. (20 Marks) Do the series converges absolutely, converges conditionally, or diverges? Solution (1) The series diverges since it is a p -series with p =1/2<1. (2) The series is an alternating series and satifies ( i ) for all n . ( ii ) . It follows from the Alternating Serire Test that converges. (3) By (1) and (2), converges conditionally. (20 Marks)Find the convergence set for .

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

View Full Document
Solution Let and . Then and Hence, we have ( i ) when | x -1|<2 (or -1< x <3), the series converges absolutely. ( ii ) When | x -1|>2 (or x <-1 or x >3), the series diverges. ( iii ) When x -1=2 (or x =3), the series becomes and diverges. ( iv ) When x -1=-2 (or x =-1), the series becomes . It follows from the Alternating Series Test that converges. Hence, the convergence set is or [-1, 3). (10 Marks)Find the Maclaurin series for and show that it represents

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: for all . (You may use for each . Solution Since , . Hence, we have where and c is some point between 0 and x . Since , we have for each . It follows that (10 Marks)The power series representation for the function begins Find the coefficient of in the series. Solution Since for , the coefficient of is (20 Marks)A function f ( x ) satisfies f (1)=3, and its first four derivatives are as follows: Unfortunately, we do not know a formula for f ( x ). ( i ) Find , the Taylor polynomial of order 3 based at a =1. ( ii ) Give a bound for , the error in Taylor's Formula with n =3. Solution ( i ) f (1)=3, , and . Hence, we have ( ii ) Since , Hence, when 1< c <1.5, we have Since , we have (20 Marks)Use a calculator to estimate using the Trapezoidal Rule with n =3 (Three decimal places are enough). Solution , a =2, b =4 and n =3. Then we have h =( b-a )/ n =(4-2)/3=2/3. , , , , • About this document . .. Next: About this document Kunquan Lan Tue Mar 7 16:47:02 EST 2000...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

Exam 4 - for all(You may use for each Solution Since Hence...

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

View Full Document
Ask a homework question - tutors are online