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.

,

,

,

,

Tue Mar 7 16:47:02 EST 2000