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

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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 .
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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
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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&lt; c &lt;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...
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This note was uploaded on 12/16/2009 for the course MATH 1014 taught by Professor Ganong during the Fall '09 term at York University.

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

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