MATH 1014.03MW Exam 2 Wednesday, February, 9 2000
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.
(c) . ¯
(c) . (d) .
Solution: (a) .
(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
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
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