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MATH 1014.03MW Exam 3 Monday, March, 6 2000


Student Number:tex2html_wrap_inline296 No.tex2html_wrap_inline298Marks tex2html_wrap_inline300

Instructions: This exam contains 6 questions and has a total of 100 marks. Show all of your work.

tex2html_wrap_inline304 (20 Marks) Do the series tex2html_wrap_inline306 converges absolutely, converges conditionally, or diverges?

Solution (1) The series tex2html_wrap_inline310 diverges since it is a p-series with p=1/2<1.

(2) The series tex2html_wrap_inline306 is an alternating series and satifies

(i) tex2html_wrap_inline322 for all n.

(ii) tex2html_wrap_inline328.

It follows from the Alternating Serire Test that tex2html_wrap_inline306 converges.

(3) By (1) and (2), tex2html_wrap_inline306 converges conditionally.

tex2html_wrap_inline340(20 Marks)Find the convergence set for tex2html_wrap_inline342.

Solution Let tex2html_wrap_inline344 and tex2html_wrap_inline346. Then tex2html_wrap_inline348 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 tex2html_wrap_inline372 and diverges.

(iv) When x-1=-2 (or x=-1), the series becomes tex2html_wrap_inline380. It follows from the Alternating Series Test that tex2html_wrap_inline380 converges.

Hence, the convergence set is tex2html_wrap_inline384 or [-1, 3).

tex2html_wrap_inline388(10 Marks)Find the Maclaurin series for tex2html_wrap_inline390 and show that it represents tex2html_wrap_inline392 for all tex2html_wrap_inline394. (You may use tex2html_wrap_inline396 for each tex2html_wrap_inline398.

Solution Since tex2html_wrap_inline400, tex2html_wrap_inline402. Hence, we have

where tex2html_wrap_inline406 and c is some point between 0 and x. Since tex2html_wrap_inline396, we have tex2html_wrap_inline416 for each tex2html_wrap_inline394. It follows that

tex2html_wrap_inline422(10 Marks)The power series representation for the function tex2html_wrap_inline424 begins

Find the coefficient of tex2html_wrap_inline428 in the series.

Solution Since tex2html_wrap_inline430 for tex2html_wrap_inline394, the coefficient of tex2html_wrap_inline428 is

tex2html_wrap_inline438(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 tex2html_wrap_inline452, the Taylor polynomial of order 3 based at a=1.

(ii) Give a bound for tex2html_wrap_inline460, the error in Taylor's Formula with n=3.

Solution (i) f(1)=3, tex2html_wrap_inline468, tex2html_wrap_inline470 and tex2html_wrap_inline472.

Hence, we have

(ii) Since tex2html_wrap_inline478, Hence, when 1<c<1.5, we have

Since tex2html_wrap_inline484, we have

tex2html_wrap_inline488(20 Marks)Use a calculator to estimate tex2html_wrap_inline490 using the Trapezoidal Rule with n=3 (Three decimal places are enough).

Solution tex2html_wrap_inline494, a=2, b=4 and n=3. Then we have h=(b-a)/n=(4-2)/3=2/3.

tex2html_wrap_inline504, tex2html_wrap_inline506

tex2html_wrap_inline508, tex2html_wrap_inline510

tex2html_wrap_inline512, tex2html_wrap_inline514

tex2html_wrap_inline516, tex2html_wrap_inline518


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Kunquan Lan
Tue Mar 7 16:47:02 EST 2000