MATH 1014.03MW Exam 3 Monday, March, 6 2000
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.
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
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
Find the coefficient of in the series.
Solution Since for , the coefficient of
(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
when 1<c<1.5, 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.