MATH 1014.03MW **Exam 2** Wednesday, February, 9 2000

NAME:

Student Number: No.Marks

**Instructions**: This exam contains 3 questions each of which has
several parts and has a total
of 100 marks. Show all of your work.

(40 Marks) Evaluate each of the following limits.

(a) (b)(

a) (b)

(c) . ¯(

c) . (d) .

Solution: (*a*)
.

(*b*)

Hence, .

(3) . =

(20 Marks)Evaluate each improper integral or show that it diverges.

(a) . (b) ,p>0. ¯(

a) . (b) ,p>0.

Solution: (*a*) *x*=0 is a singularity. Let . Then

. Hence, we have

This implies .

(*b*) Let . Then *x*=1 is a singularity. Let . Then

We consider the following cases:

(*i*) When 1-*p*>0,
and
converges.

(*ii*) When 1-*p*<0, and diverges.

(*iii*) When *p*=1, . Hence, .
In this case,
diverges.

(40 Marks)Determine convergence or divergence for each of the following series.

(a) . ¯(

a) . (b) .

(c) . (d)(

c) . (d) (p>0,x>0).

Solution: (*a*) (Method 1: Bounded Sum Test) Let
. Note
that for . Hence, for ,

Hence, is bounded and converges.

There is another way to show is bounded. Note that for . Hence, for ,

(Method 2: Ordinary Comparison Test) Let , Then for . Since converges, it follows that converges.

Another way: for . Since converges, it follows that converges.

(Method 3: Ratio Test) . It follows from Ratio Test that the series converges.

(Method 4: Root Test) Noting that , we have . It follows that converges.

(*b*) Since and
diverges,
it follows from Limit Comparison Test that
diverges.

(*c*) Let . Then
is continuous since is
continuous on and for . Also,
, so *f* is
decreasing. Moreover,

Hence, and converges. It follows from Integral Test that converges.

(*d*) Let . Then .

Hence, . By Ratio Test, we have

(*i*) If *x*<1,
converges.

(*ii*) If *x*>1,
diverges.

When *x*=1, the series becomes
. It is known that when
*p*>1, converges and
when ,
diverges.

Wed Feb 9 09:35:54 EST 2000