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.