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 .

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 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.

,

,

,

,